113 research outputs found
DTM-based Filtrations
Despite strong stability properties, the persistent homology of filtrations
classically used in Topological Data Analysis, such as, e.g. the Cech or
Vietoris-Rips filtrations, are very sensitive to the presence of outliers in
the data from which they are computed. In this paper, we introduce and study a
new family of filtrations, the DTM-filtrations, built on top of point clouds in
the Euclidean space which are more robust to noise and outliers. The approach
adopted in this work relies on the notion of distance-to-measure functions, and
extends some previous work on the approximation of such functions.Comment: Abel Symposia, Springer, In press, Topological Data Analysi
DTM-Based Filtrations
Despite strong stability properties, the persistent homology of filtrations classically used in Topological Data Analysis, such as, e.g. the Cech or Vietoris-Rips filtrations, are very sensitive to the presence of outliers in the data from which they are computed. In this paper, we introduce and study a new family of filtrations, the DTM-filtrations, built on top of point clouds in the Euclidean space which are more robust to noise and outliers. The approach adopted in this work relies on the notion of distance-to-measure functions and extends some previous work on the approximation of such functions
DTM-based Filtrations
International audienceDespite strong stability properties, the persistent homology of filtrations classically used in Topological Data Analysis, such as, e.g. the Cech or Vietoris-Rips filtrations, are very sensitive to the presence of outliers in the data from which they are computed. In this paper, we introduce and study a new family of filtrations, the DTM-filtrations, built on top of point clouds in the Euclidean space which are more robust to noise and outliers. The approach adopted in this work relies on the notion of distance-to-measure functions, and extends some previous work on the approximation of such functions
The Persistence of Large Scale Structures I: Primordial non-Gaussianity
We develop an analysis pipeline for characterizing the topology of large
scale structure and extracting cosmological constraints based on persistent
homology. Persistent homology is a technique from topological data analysis
that quantifies the multiscale topology of a data set, in our context unifying
the contributions of clusters, filament loops, and cosmic voids to cosmological
constraints. We describe how this method captures the imprint of primordial
local non-Gaussianity on the late-time distribution of dark matter halos, using
a set of N-body simulations as a proxy for real data analysis. For our best
single statistic, running the pipeline on several cubic volumes of size
, we detect at
confidence on of the volumes. Additionally we test our ability to
resolve degeneracies between the topological signature of and variation of and argue that correctly identifying nonzero
in this case is possible via an optimal template method.
Our method relies on information living at Mpc/h, a
complementary scale with respect to commonly used methods such as the
scale-dependent bias in the halo/galaxy power spectrum. Therefore, while still
requiring a large volume, our method does not require sampling long-wavelength
modes to constrain primordial non-Gaussianity. Moreover, our statistics are
interpretable: we are able to reproduce previous results in certain limits and
we make new predictions for unexplored observables, such as filament loops
formed by dark matter halos in a simulation box.Comment: 33+11 pages, 19 figures, code available at
https://gitlab.com/mbiagetti/persistent_homology_ls
Recovering the homology of immersed manifolds
Given a sample of an abstract manifold immersed in some Euclidean space, we
describe a way to recover the singular homology of the original manifold. It
consists in estimating its tangent bundle---seen as subset of another Euclidean
space---in a measure theoretic point of view, and in applying measure-based
filtrations for persistent homology. The construction we propose is consistent
and stable, and does not involve the knowledge of the dimension of the
manifold.In order to obtain quantitative results, we introduce the normal
reach, which is a notion of reach suitable for an immersed manifold
Adaptive Topological Feature via Persistent Homology: Filtration Learning for Point Clouds
Machine learning for point clouds has been attracting much attention, with
many applications in various fields, such as shape recognition and material
science. To enhance the accuracy of such machine learning methods, it is known
to be effective to incorporate global topological features, which are typically
extracted by persistent homology. In the calculation of persistent homology for
a point cloud, we need to choose a filtration for the point clouds, an
increasing sequence of spaces. Because the performance of machine learning
methods combined with persistent homology is highly affected by the choice of a
filtration, we need to tune it depending on data and tasks. In this paper, we
propose a framework that learns a filtration adaptively with the use of neural
networks. In order to make the resulting persistent homology
isometry-invariant, we develop a neural network architecture with such
invariance. Additionally, we theoretically show a finite-dimensional
approximation result that justifies our architecture. Experimental results
demonstrated the efficacy of our framework in several classification tasks.Comment: 17 pages with 4 figure
Sparse Higher Order ?ech Filtrations
For a finite set of balls of radius r, the k-fold cover is the space covered by at least k balls. Fixing the ball centers and varying the radius, we obtain a nested sequence of spaces that is called the k-fold filtration of the centers. For k = 1, the construction is the union-of-balls filtration that is popular in topological data analysis. For larger k, it yields a cleaner shape reconstruction in the presence of outliers. We contribute a sparsification algorithm to approximate the topology of the k-fold filtration. Our method is a combination and adaptation of several techniques from the well-studied case k = 1, resulting in a sparsification of linear size that can be computed in expected near-linear time with respect to the number of input points
Detection of Small Holes by the Scale-Invariant Robust Density-Aware Distance (RDAD) Filtration
A novel topological-data-analytical (TDA) method is proposed to distinguish,
from noise, small holes surrounded by high-density regions of a probability
density function whose mass is concentrated near a manifold (or more generally,
a CW complex) embedded in a high-dimensional Euclidean space. The proposed
method is robust against additive noise and outliers. In particular, sample
points are allowed to be perturbed away from the manifold. Traditional TDA
tools, like those based on the distance filtration, often struggle to
distinguish small features from noise, because of their short persistence. An
alternative filtration, called Robust Density-Aware Distance (RDAD) filtration,
is proposed to prolong the persistence of small holes surrounded by
high-density regions. This is achieved by weighting the distance function by
the density in the sense of Bell et al. Distance-to-measure is incorporated to
enhance stability and mitigate noise due to the density estimation. The utility
of the proposed filtration in identifying small holes, as well as its
robustness against noise, are illustrated through an analytical example and
extensive numerical experiments. Basic mathematical properties of the proposed
filtration are proven.Comment: 47 pages, 60 figures, GitHub repo: https://github.com/c-siu/RDA
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