8,861 research outputs found
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
Computing the homology of basic semialgebraic sets in weak exponential time
We describe and analyze an algorithm for computing the homology (Betti
numbers and torsion coefficients) of basic semialgebraic sets which works in
weak exponential time. That is, out of a set of exponentially small measure in
the space of data the cost of the algorithm is exponential in the size of the
data. All algorithms previously proposed for this problem have a complexity
which is doubly exponential (and this is so for almost all data)
Hyperplane Neural Codes and the Polar Complex
Hyperplane codes are a class of convex codes that arise as the output of a
one layer feed-forward neural network. Here we establish several natural
properties of stable hyperplane codes in terms of the {\it polar complex} of
the code, a simplicial complex associated to any combinatorial code. We prove
that the polar complex of a stable hyperplane code is shellable and show that
most currently known properties of the hyperplane codes follow from the
shellability of the appropriate polar complex.Comment: 23 pages, 5 figures. To appear in Proceedings of the Abel Symposiu
Towards Stratification Learning through Homology Inference
A topological approach to stratification learning is developed for point
cloud data drawn from a stratified space. Given such data, our objective is to
infer which points belong to the same strata. First we define a multi-scale
notion of a stratified space, giving a stratification for each radius level. We
then use methods derived from kernel and cokernel persistent homology to
cluster the data points into different strata, and we prove a result which
guarantees the correctness of our clustering, given certain topological
conditions; some geometric intuition for these topological conditions is also
provided. Our correctness result is then given a probabilistic flavor: we give
bounds on the minimum number of sample points required to infer, with
probability, which points belong to the same strata. Finally, we give an
explicit algorithm for the clustering, prove its correctness, and apply it to
some simulated data.Comment: 48 page
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
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