3,067 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
Approximating Loops in a Shortest Homology Basis from Point Data
Inference of topological and geometric attributes of a hidden manifold from
its point data is a fundamental problem arising in many scientific studies and
engineering applications. In this paper we present an algorithm to compute a
set of loops from a point data that presumably sample a smooth manifold
. These loops approximate a {\em shortest} basis of the
one dimensional homology group over coefficients in finite field
. Previous results addressed the issue of computing the rank of
the homology groups from point data, but there is no result on approximating
the shortest basis of a manifold from its point sample. In arriving our result,
we also present a polynomial time algorithm for computing a shortest basis of
for any finite {\em simplicial complex} whose edges have
non-negative weights
Topological analysis of scalar fields with outliers
Given a real-valued function defined over a manifold embedded in
, we are interested in recovering structural information about
from the sole information of its values on a finite sample . Existing
methods provide approximation to the persistence diagram of when geometric
noise and functional noise are bounded. However, they fail in the presence of
aberrant values, also called outliers, both in theory and practice.
We propose a new algorithm that deals with outliers. We handle aberrant
functional values with a method inspired from the k-nearest neighbors
regression and the local median filtering, while the geometric outliers are
handled using the distance to a measure. Combined with topological results on
nested filtrations, our algorithm performs robust topological analysis of
scalar fields in a wider range of noise models than handled by current methods.
We provide theoretical guarantees and experimental results on the quality of
our approximation of the sampled scalar field
Approximating Persistent Homology in Euclidean Space Through Collapses
The \v{C}ech complex is one of the most widely used tools in applied
algebraic topology. Unfortunately, due to the inclusive nature of the \v{C}ech
filtration, the number of simplices grows exponentially in the number of input
points. A practical consequence is that computations may have to terminate at
smaller scales than what the application calls for.
In this paper we propose two methods to approximate the \v{C}ech persistence
module. Both are constructed on the level of spaces, i.e. as sequences of
simplicial complexes induced by nerves. We also show how the bottleneck
distance between such persistence modules can be understood by how tightly they
are sandwiched on the level of spaces. In turn, this implies the correctness of
our approximation methods.
Finally, we implement our methods and apply them to some example point clouds
in Euclidean space
Random curves on surfaces induced from the Laplacian determinant
We define natural probability measures on cycle-rooted spanning forests
(CRSFs) on graphs embedded on a surface with a Riemannian metric. These
measures arise from the Laplacian determinant and depend on the choice of a
unitary connection on the tangent bundle to the surface.
We show that, for a sequence of graphs conformally approximating the
surface, the measures on CRSFs of converge and give a limiting
probability measure on finite multicurves (finite collections of pairwise
disjoint simple closed curves) on the surface, independent of the approximating
sequence.
Wilson's algorithm for generating spanning trees on a graph generalizes to a
cycle-popping algorithm for generating CRSFs for a general family of weights on
the cycles. We use this to sample the above measures. The sampling algorithm,
which relates these measures to the loop-erased random walk, is also used to
prove tightness of the sequence of measures, a key step in the proof of their
convergence.
We set the framework for the study of these probability measures and their
scaling limits and state some of their properties
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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