331 research outputs found
Approximating Local Homology from Samples
Recently, multi-scale notions of local homology (a variant of persistent
homology) have been used to study the local structure of spaces around a given
point from a point cloud sample. Current reconstruction guarantees rely on
constructing embedded complexes which become difficult in high dimensions. We
show that the persistence diagrams used for estimating local homology, can be
approximated using families of Vietoris-Rips complexes, whose simple
constructions are robust in any dimension. To the best of our knowledge, our
results, for the first time, make applications based on local homology, such as
stratification learning, feasible in high dimensions.Comment: 23 pages, 14 figure
On Vietoris-Rips complexes of ellipses
For a metric space and a scale parameter, the Vietoris-Rips complex
(resp. ) has as its vertex set, and a finite
subset as a simplex whenever the diameter of is
less than (resp. at most ). Though Vietoris-Rips complexes have been
studied at small choices of scale by Hausmann and Latschev, they are not
well-understood at larger scale parameters. In this paper we investigate the
homotopy types of Vietoris-Rips complexes of ellipses of small eccentricity, meaning
. Indeed, we show there are constants such that for
all , we have and , though only one of the two-spheres in is
persistent. Furthermore, we show that for any scale parameter ,
there are arbitrarily dense subsets of the ellipse such that the Vietoris-Rips
complex of the subset is not homotopy equivalent to the Vietoris-Rips complex
of the entire ellipse. As our main tool we link these homotopy types to the
structure of infinite cyclic graphs
Persistent Homology analysis of Phase Transitions
Persistent homology analysis, a recently developed computational method in
algebraic topology, is applied to the study of the phase transitions undergone
by the so-called XY-mean field model and by the phi^4 lattice model,
respectively. For both models the relationship between phase transitions and
the topological properties of certain submanifolds of configuration space are
exactly known. It turns out that these a-priori known facts are clearly
retrieved by persistent homology analysis of dynamically sampled submanifolds
of configuration space.Comment: 10 pages; 10 figure
The Vietoris-Rips complexes of a circle
Given a metric space X and a distance threshold r>0, the Vietoris-Rips
simplicial complex has as its simplices the finite subsets of X of diameter
less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian
manifold and r is sufficiently small, then the Vietoris-Rips complex is
homotopy equivalent to the original manifold. Little is known about the
behavior of Vietoris-Rips complexes for larger values of r, even though these
complexes arise naturally in applications using persistent homology. We show
that as r increases, the Vietoris-Rips complex of the circle obtains the
homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ...,
until finally it is contractible. As our main tool we introduce a directed
graph invariant, the winding fraction, which in some sense is dual to the
circular chromatic number. Using the winding fraction we classify the homotopy
types of the Vietoris-Rips complex of an arbitrary (possibly infinite) subset
of the circle, and we study the expected homotopy type of the Vietoris-Rips
complex of a uniformly random sample from the circle. Moreover, we show that as
the distance parameter increases, the ambient Cech complex of the circle also
obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the
7-sphere, ..., until finally it is contractible.Comment: Final versio
Dimension Detection with Local Homology
Detecting the dimension of a hidden manifold from a point sample has become
an important problem in the current data-driven era. Indeed, estimating the
shape dimension is often the first step in studying the processes or phenomena
associated to the data. Among the many dimension detection algorithms proposed
in various fields, a few can provide theoretical guarantee on the correctness
of the estimated dimension. However, the correctness usually requires certain
regularity of the input: the input points are either uniformly randomly sampled
in a statistical setting, or they form the so-called
-sample which can be neither too dense nor too sparse.
Here, we propose a purely topological technique to detect dimensions. Our
algorithm is provably correct and works under a more relaxed sampling
condition: we do not require uniformity, and we also allow Hausdorff noise. Our
approach detects dimension by determining local homology. The computation of
this topological structure is much less sensitive to the local distribution of
points, which leads to the relaxation of the sampling conditions. Furthermore,
by leveraging various developments in computational topology, we show that this
local homology at a point can be computed \emph{exactly} for manifolds
using Vietoris-Rips complexes whose vertices are confined within a local
neighborhood of . We implement our algorithm and demonstrate the accuracy
and robustness of our method using both synthetic and real data sets
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