9,129 research outputs found
A Stable Multi-Scale Kernel for Topological Machine Learning
Topological data analysis offers a rich source of valuable information to
study vision problems. Yet, so far we lack a theoretically sound connection to
popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In
this work, we establish such a connection by designing a multi-scale kernel for
persistence diagrams, a stable summary representation of topological features
in data. We show that this kernel is positive definite and prove its stability
with respect to the 1-Wasserstein distance. Experiments on two benchmark
datasets for 3D shape classification/retrieval and texture recognition show
considerable performance gains of the proposed method compared to an
alternative approach that is based on the recently introduced persistence
landscapes
Geometry Helps to Compare Persistence Diagrams
Exploiting geometric structure to improve the asymptotic complexity of
discrete assignment problems is a well-studied subject. In contrast, the
practical advantages of using geometry for such problems have not been
explored. We implement geometric variants of the Hopcroft--Karp algorithm for
bottleneck matching (based on previous work by Efrat el al.) and of the auction
algorithm by Bertsekas for Wasserstein distance computation. Both
implementations use k-d trees to replace a linear scan with a geometric
proximity query. Our interest in this problem stems from the desire to compute
distances between persistence diagrams, a problem that comes up frequently in
topological data analysis. We show that our geometric matching algorithms lead
to a substantial performance gain, both in running time and in memory
consumption, over their purely combinatorial counterparts. Moreover, our
implementation significantly outperforms the only other implementation
available for comparing persistence diagrams.Comment: 20 pages, 10 figures; extended version of paper published in ALENEX
201
Subsampling Methods for Persistent Homology
Persistent homology is a multiscale method for analyzing the shape of sets
and functions from point cloud data arising from an unknown distribution
supported on those sets. When the size of the sample is large, direct
computation of the persistent homology is prohibitive due to the combinatorial
nature of the existing algorithms. We propose to compute the persistent
homology of several subsamples of the data and then combine the resulting
estimates. We study the risk of two estimators and we prove that the
subsampling approach carries stable topological information while achieving a
great reduction in computational complexity
Progressive Wasserstein Barycenters of Persistence Diagrams
This paper presents an efficient algorithm for the progressive approximation
of Wasserstein barycenters of persistence diagrams, with applications to the
visual analysis of ensemble data. Given a set of scalar fields, our approach
enables the computation of a persistence diagram which is representative of the
set, and which visually conveys the number, data ranges and saliences of the
main features of interest found in the set. Such representative diagrams are
obtained by computing explicitly the discrete Wasserstein barycenter of the set
of persistence diagrams, a notoriously computationally intensive task. In
particular, we revisit efficient algorithms for Wasserstein distance
approximation [12,51] to extend previous work on barycenter estimation [94]. We
present a new fast algorithm, which progressively approximates the barycenter
by iteratively increasing the computation accuracy as well as the number of
persistent features in the output diagram. Such a progressivity drastically
improves convergence in practice and allows to design an interruptible
algorithm, capable of respecting computation time constraints. This enables the
approximation of Wasserstein barycenters within interactive times. We present
an application to ensemble clustering where we revisit the k-means algorithm to
exploit our barycenters and compute, within execution time constraints,
meaningful clusters of ensemble data along with their barycenter diagram.
Extensive experiments on synthetic and real-life data sets report that our
algorithm converges to barycenters that are qualitatively meaningful with
regard to the applications, and quantitatively comparable to previous
techniques, while offering an order of magnitude speedup when run until
convergence (without time constraint). Our algorithm can be trivially
parallelized to provide additional speedups in practice on standard
workstations. [...
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