381 research outputs found
Topological analysis of scalar fields with outliers
Given a real-valued function defined over a manifold embedded in
, we are interested in recovering structural information about
from the sole information of its values on a finite sample . Existing
methods provide approximation to the persistence diagram of when geometric
noise and functional noise are bounded. However, they fail in the presence of
aberrant values, also called outliers, both in theory and practice.
We propose a new algorithm that deals with outliers. We handle aberrant
functional values with a method inspired from the k-nearest neighbors
regression and the local median filtering, while the geometric outliers are
handled using the distance to a measure. Combined with topological results on
nested filtrations, our algorithm performs robust topological analysis of
scalar fields in a wider range of noise models than handled by current methods.
We provide theoretical guarantees and experimental results on the quality of
our approximation of the sampled scalar field
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
Optimal rates of convergence for persistence diagrams in Topological Data Analysis
Computational topology has recently known an important development toward
data analysis, giving birth to the field of topological data analysis.
Topological persistence, or persistent homology, appears as a fundamental tool
in this field. In this paper, we study topological persistence in general
metric spaces, with a statistical approach. We show that the use of persistent
homology can be naturally considered in general statistical frameworks and
persistence diagrams can be used as statistics with interesting convergence
properties. Some numerical experiments are performed in various contexts to
illustrate our results
Radiative Contributions to the Effective Action of Self-Interacting Scalar Field on a Manifold with Boundary
The effect of quantum corrections to a conformally invariant field theory for
a self-interacting scalar field on a curved manifold with boundary is
considered. The analysis is most easily performed in a space of constant
curvature the boundary of which is characterised by constant extrinsic
curvature. An extension of the spherical formulation in the presence of a
boundary is attained through use of the method of images. Contrary to the
consolidated vanishing effect in maximally symmetric space-times the
contribution of the massless "tadpole" diagram no longer vanishes in
dimensional regularisation. As a result, conformal invariance is broken due to
boundary-related vacuum contributions. The evaluation of one-loop contributions
to the two-point function suggests an extension, in the presence of matter
couplings, of the simultaneous volume and boundary renormalisation in the
effective action.Comment: 14 pages, 1 figure. Additional references and minor elucidating
remarks added. To appear in Classical and Quantum Gravit
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