1,823 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
Pyrolysis of asphaltenes and biomarkers for the fingerprinting of the _Amoco Cadiz_ oil spill after 23 years
The chemical composition of the petroleum products accidently or deliberately released in the environment varies considerably with time under the action of biological (biodegradation) and physico-chemical (photo-oxidation) processes. It becomes more and more difficult to trace the origin of the oil spilled. A technique widely used for monitoring ancient oil pollutions is the study of oil biomarkers like terpanes and steranes^1,2^. Here we show that the geochemical technique of asphaltenes pyrolysis can be successfully applied to environmental samples. This method allows the reconstitution of the original oil from the asphaltenes fraction of severely degraded oil residues. We applied the two techniques: biomarkers analysis and pyrolysis of asphaltenes to the long-term characterisation of the _Amoco Cadiz_ oil 23 years after the spill in the salt marshes of Ile Grande, Northern Brittany, France. The results show that the oil reached the ultimate degradation stage. The total biodegradation rate was 60% relatively to initial oil. The asphaltenes pyrolysis generated a gas-chromatographic profile very similar to the original _Amoco Cadiz_ oil. In the biomarkers fraction, gas chromatographic/mass spectrometric (GC/MS) analyses demonstrated that terpanes were conserved whereas steranes were partly degraded. We also showed that the class of seco-hopanes biomarkers are conserved and can be used in the long term monitoring of oil pollutions
Barcode Embeddings for Metric Graphs
Stable topological invariants are a cornerstone of persistence theory and
applied topology, but their discriminative properties are often
poorly-understood. In this paper we study a rich homology-based invariant first
defined by Dey, Shi, and Wang, which we think of as embedding a metric graph in
the barcode space. We prove that this invariant is locally injective on the
space of metric graphs and globally injective on a GH-dense subset. Moreover,
we show that is globally injective on a full measure subset of metric graphs,
in the appropriate sense.Comment: The newest draft clarifies the proofs in Sections 7 and 8, and
provides improved figures therein. It also includes a results section in the
introductio
Local Equivalence and Intrinsic Metrics between Reeb Graphs
As graphical summaries for topological spaces and maps, Reeb graphs are
common objects in the computer graphics or topological data analysis
literature. Defining good metrics between these objects has become an important
question for applications, where it matters to quantify the extent by which two
given Reeb graphs differ. Recent contributions emphasize this aspect, proposing
novel distances such as {\em functional distortion} or {\em interleaving} that
are provably more discriminative than the so-called {\em bottleneck distance},
being true metrics whereas the latter is only a pseudo-metric. Their main
drawback compared to the bottleneck distance is to be comparatively hard (if at
all possible) to evaluate. Here we take the opposite view on the problem and
show that the bottleneck distance is in fact good enough {\em locally}, in the
sense that it is able to discriminate a Reeb graph from any other Reeb graph in
a small enough neighborhood, as efficiently as the other metrics do. This
suggests considering the {\em intrinsic metrics} induced by these distances,
which turn out to be all {\em globally} equivalent. This novel viewpoint on the
study of Reeb graphs has a potential impact on applications, where one may not
only be interested in discriminating between data but also in interpolating
between them
Sliced Wasserstein Kernel for Persistence Diagrams
Persistence diagrams (PDs) play a key role in topological data analysis
(TDA), in which they are routinely used to describe topological properties of
complicated shapes. PDs enjoy strong stability properties and have proven their
utility in various learning contexts. They do not, however, live in a space
naturally endowed with a Hilbert structure and are usually compared with
specific distances, such as the bottleneck distance. To incorporate PDs in a
learning pipeline, several kernels have been proposed for PDs with a strong
emphasis on the stability of the RKHS distance w.r.t. perturbations of the PDs.
In this article, we use the Sliced Wasserstein approximation SW of the
Wasserstein distance to define a new kernel for PDs, which is not only provably
stable but also provably discriminative (depending on the number of points in
the PDs) w.r.t. the Wasserstein distance between PDs. We also demonstrate
its practicality, by developing an approximation technique to reduce kernel
computation time, and show that our proposal compares favorably to existing
kernels for PDs on several benchmarks.Comment: Minor modification
Statistical Analysis and Parameter Selection for Mapper
In this article, we study the question of the statistical convergence of the
1-dimensional Mapper to its continuous analogue, the Reeb graph. We show that
the Mapper is an optimal estimator of the Reeb graph, which gives, as a
byproduct, a method to automatically tune its parameters and compute confidence
regions on its topological features, such as its loops and flares. This allows
to circumvent the issue of testing a large grid of parameters and keeping the
most stable ones in the brute-force setting, which is widely used in
visualization, clustering and feature selection with the Mapper.Comment: Minor modification
Persistence stability for geometric complexes
In this paper we study the properties of the homology of different geometric
filtered complexes (such as Vietoris-Rips, Cech and witness complexes) built on
top of precompact spaces. Using recent developments in the theory of
topological persistence we provide simple and natural proofs of the stability
of the persistent homology of such complexes with respect to the
Gromov--Hausdorff distance. We also exhibit a few noteworthy properties of the
homology of the Rips and Cech complexes built on top of compact spaces.Comment: We include a discussion of ambient Cech complexes and a new class of
examples called Dowker complexe
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