1,383 research outputs found
Persistence Bag-of-Words for Topological Data Analysis
Persistent homology (PH) is a rigorous mathematical theory that provides a
robust descriptor of data in the form of persistence diagrams (PDs). PDs
exhibit, however, complex structure and are difficult to integrate in today's
machine learning workflows. This paper introduces persistence bag-of-words: a
novel and stable vectorized representation of PDs that enables the seamless
integration with machine learning. Comprehensive experiments show that the new
representation achieves state-of-the-art performance and beyond in much less
time than alternative approaches.Comment: Accepted for the Twenty-Eight International Joint Conference on
Artificial Intelligence (IJCAI-19). arXiv admin note: substantial text
overlap with arXiv:1802.0485
Topological descriptors for 3D surface analysis
We investigate topological descriptors for 3D surface analysis, i.e. the
classification of surfaces according to their geometric fine structure. On a
dataset of high-resolution 3D surface reconstructions we compute persistence
diagrams for a 2D cubical filtration. In the next step we investigate different
topological descriptors and measure their ability to discriminate structurally
different 3D surface patches. We evaluate their sensitivity to different
parameters and compare the performance of the resulting topological descriptors
to alternative (non-topological) descriptors. We present a comprehensive
evaluation that shows that topological descriptors are (i) robust, (ii) yield
state-of-the-art performance for the task of 3D surface analysis and (iii)
improve classification performance when combined with non-topological
descriptors.Comment: 12 pages, 3 figures, CTIC 201
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
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