15,621 research outputs found
Computational Topology Techniques for Characterizing Time-Series Data
Topological data analysis (TDA), while abstract, allows a characterization of
time-series data obtained from nonlinear and complex dynamical systems. Though
it is surprising that such an abstract measure of structure - counting pieces
and holes - could be useful for real-world data, TDA lets us compare different
systems, and even do membership testing or change-point detection. However, TDA
is computationally expensive and involves a number of free parameters. This
complexity can be obviated by coarse-graining, using a construct called the
witness complex. The parametric dependence gives rise to the concept of
persistent homology: how shape changes with scale. Its results allow us to
distinguish time-series data from different systems - e.g., the same note
played on different musical instruments.Comment: 12 pages, 6 Figures, 1 Table, The Sixteenth International Symposium
on Intelligent Data Analysis (IDA 2017
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
Manifold Reconstruction in Arbitrary Dimensions using Witness Complexes
A preliminary version of this paper has been presented at the ACM Symp. on Computational Geometry 2007International audienceIt is a well-established fact that the witness complex is closely related to the restricted Delaunay triangulation in low dimensions. Specifically, it has been proved that the witness complex coincides with the restricted Delaunay triangulation on curves, and is still a subset of it on surfaces, under mild sampling conditions. In this paper, we prove that these results do not extend to higher-dimensional manifolds, even under strong sampling conditions such as uniform point density. On the positive side, we show how the sets of witnesses and landmarks can be enriched, so that the nice relations that exist between restricted Delaunay triangulation and witness complex hold on higher-dimensional manifolds as well. We derive from our structural results an algorithm that reconstructs manifolds of any arbitrary dimension or co-dimension at different scales. The algorithm combines a farthest-point refinement scheme with a vertex pumping strategy. It is very simple conceptually, and it does not require the input point sample to be sparse. Its running time is bounded by c(d)n2, where n is the size of the input point cloud, and c(d) is a constant depending solely (yet exponentially) on the dimension d of the ambient space. Although this running time makes our reconstruction algorithm rather theoretical, recent work has shown that a variant of our approach can be made tractable in arbitrary dimensions, by building upon the results of this paper
Only distances are required to reconstruct submanifolds
In this paper, we give the first algorithm that outputs a faithful
reconstruction of a submanifold of Euclidean space without maintaining or even
constructing complicated data structures such as Voronoi diagrams or Delaunay
complexes. Our algorithm uses the witness complex and relies on the stability
of power protection, a notion introduced in this paper. The complexity of the
algorithm depends exponentially on the intrinsic dimension of the manifold,
rather than the dimension of ambient space, and linearly on the dimension of
the ambient space. Another interesting feature of this work is that no explicit
coordinates of the points in the point sample is needed. The algorithm only
needs the distance matrix as input, i.e., only distance between points in the
point sample as input.Comment: Major revision, 16 figures, 47 page
Witness structures and immediate snapshot complexes
In this paper we introduce and study a new family of combinatorial simplicial
complexes, which we call immediate snapshot complexes. Our construction and
terminology is strongly motivated by theoretical distributed computing, as
these complexes are combinatorial models of the standard protocol complexes
associated to immediate snapshot read/write shared memory communication model.
In order to define the immediate snapshot complexes we need a new combinatorial
object, which we call a witness structure. These objects are indexing the
simplices in the immediate snapshot complexes, while a special operation on
them, called ghosting, describes the combinatorics of taking simplicial
boundary. In general, we develop the theory of witness structures and use it to
prove several combinatorial as well as topological properties of the immediate
snapshot complexes.Comment: full paper version of the 1st part of the preprint arXiv:1402.4707;
to appear in DMTC
A Growing Self-Organizing Network for Reconstructing Curves and Surfaces
Self-organizing networks such as Neural Gas, Growing Neural Gas and many
others have been adopted in actual applications for both dimensionality
reduction and manifold learning. Typically, in these applications, the
structure of the adapted network yields a good estimate of the topology of the
unknown subspace from where the input data points are sampled. The approach
presented here takes a different perspective, namely by assuming that the input
space is a manifold of known dimension. In return, the new type of growing
self-organizing network presented gains the ability to adapt itself in way that
may guarantee the effective and stable recovery of the exact topological
structure of the input manifold
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
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