4,966 research outputs found
Towards Persistence-Based Reconstruction in Euclidean Spaces
Manifold reconstruction has been extensively studied for the last decade or
so, especially in two and three dimensions. Recently, significant improvements
were made in higher dimensions, leading to new methods to reconstruct large
classes of compact subsets of Euclidean space . However, the complexities
of these methods scale up exponentially with d, which makes them impractical in
medium or high dimensions, even for handling low-dimensional submanifolds. In
this paper, we introduce a novel approach that stands in-between classical
reconstruction and topological estimation, and whose complexity scales up with
the intrinsic dimension of the data. Specifically, when the data points are
sufficiently densely sampled from a smooth -submanifold of , our
method retrieves the homology of the submanifold in time at most ,
where is the size of the input and is a constant depending solely on
. It can also provably well handle a wide range of compact subsets of
, though with worse complexities. Along the way to proving the
correctness of our algorithm, we obtain new results on \v{C}ech, Rips, and
witness complex filtrations in Euclidean spaces
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
Exact Computation of a Manifold Metric, via Lipschitz Embeddings and Shortest Paths on a Graph
Data-sensitive metrics adapt distances locally based the density of data
points with the goal of aligning distances and some notion of similarity. In
this paper, we give the first exact algorithm for computing a data-sensitive
metric called the nearest neighbor metric. In fact, we prove the surprising
result that a previously published -approximation is an exact algorithm.
The nearest neighbor metric can be viewed as a special case of a
density-based distance used in machine learning, or it can be seen as an
example of a manifold metric. Previous computational research on such metrics
despaired of computing exact distances on account of the apparent difficulty of
minimizing over all continuous paths between a pair of points. We leverage the
exact computation of the nearest neighbor metric to compute sparse spanners and
persistent homology. We also explore the behavior of the metric built from
point sets drawn from an underlying distribution and consider the more general
case of inputs that are finite collections of path-connected compact sets.
The main results connect several classical theories such as the conformal
change of Riemannian metrics, the theory of positive definite functions of
Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop
novel proof techniques based on the combination of screw functions and
Lipschitz extensions that may be of independent interest.Comment: 15 page
On the Reconstruction of Geodesic Subspaces of
We consider the topological and geometric reconstruction of a geodesic
subspace of both from the \v{C}ech and Vietoris-Rips filtrations
on a finite, Hausdorff-close, Euclidean sample. Our reconstruction technique
leverages the intrinsic length metric induced by the geodesics on the subspace.
We consider the distortion and convexity radius as our sampling parameters for
a successful reconstruction. For a geodesic subspace with finite distortion and
positive convexity radius, we guarantee a correct computation of its homotopy
and homology groups from the sample. For geodesic subspaces of ,
we also devise an algorithm to output a homotopy equivalent geometric complex
that has a very small Hausdorff distance to the unknown shape of interest
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 exploration of artificial neuronal network dynamics
One of the paramount challenges in neuroscience is to understand the dynamics
of individual neurons and how they give rise to network dynamics when
interconnected. Historically, researchers have resorted to graph theory,
statistics, and statistical mechanics to describe the spatiotemporal structure
of such network dynamics. Our novel approach employs tools from algebraic
topology to characterize the global properties of network structure and
dynamics.
We propose a method based on persistent homology to automatically classify
network dynamics using topological features of spaces built from various
spike-train distances. We investigate the efficacy of our method by simulating
activity in three small artificial neural networks with different sets of
parameters, giving rise to dynamics that can be classified into four regimes.
We then compute three measures of spike train similarity and use persistent
homology to extract topological features that are fundamentally different from
those used in traditional methods. Our results show that a machine learning
classifier trained on these features can accurately predict the regime of the
network it was trained on and also generalize to other networks that were not
presented during training. Moreover, we demonstrate that using features
extracted from multiple spike-train distances systematically improves the
performance of our method
Geometric Inference on Kernel Density Estimates
We show that geometric inference of a point cloud can be calculated by
examining its kernel density estimate with a Gaussian kernel. This allows one
to consider kernel density estimates, which are robust to spatial noise,
subsampling, and approximate computation in comparison to raw point sets. This
is achieved by examining the sublevel sets of the kernel distance, which
isomorphically map to superlevel sets of the kernel density estimate. We prove
new properties about the kernel distance, demonstrating stability results and
allowing it to inherit reconstruction results from recent advances in
distance-based topological reconstruction. Moreover, we provide an algorithm to
estimate its topology using weighted Vietoris-Rips complexes.Comment: To appear in SoCG 2015. 36 pages, 5 figure
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