24 research outputs found
Polynomial-Sized Topological Approximations Using The Permutahedron
Classical methods to model topological properties of point clouds, such as
the Vietoris-Rips complex, suffer from the combinatorial explosion of complex
sizes. We propose a novel technique to approximate a multi-scale filtration of
the Rips complex with improved bounds for size: precisely, for points in
, we obtain a -approximation with at most simplices of dimension or lower. In conjunction with dimension
reduction techniques, our approach yields a -approximation of size for Rips filtrations on arbitrary metric
spaces. This result stems from high-dimensional lattice geometry and exploits
properties of the permutahedral lattice, a well-studied structure in discrete
geometry.
Building on the same geometric concept, we also present a lower bound result
on the size of an approximate filtration: we construct a point set for which
every -approximation of the \v{C}ech filtration has to contain
features, provided that for .Comment: 24 pages, 1 figur
Approximation algorithms for Vietoris-Rips and Čech filtrations
Persistent Homology is a tool to analyze and visualize the shape of data from a topological viewpoint. It computes persistence, which summarizes the evolution of topological and geometric information about metric spaces over multiple scales of distances. While computing persistence is quite efficient for low-dimensional topological features, it becomes overwhelmingly expensive for medium to high-dimensional features. In this thesis, we attack this computational problem from several different angles. We present efficient techniques to approximate the persistence of metric spaces. Three of our methods are tailored towards general point clouds in Euclidean spaces. We make use of high dimensional lattice geometry to reduce the cost of the approximations. In particular, we discover several properties of the Permutahedral lattice, whose Voronoi cell is well-known for its combinatorial properties. The last method is suitable for point clouds with low intrinsic dimension, where we exploit the structural properties of the point set to tame the complexity. In some cases, we achieve a reduction in size complexity by trading off the quality of the approximation. Two of our methods work particularly well in conjunction with dimension-reduction techniques: we arrive at the first approximation schemes whose complexities are only polynomial in the size of the point cloud, and independent of the ambient dimension. On the other hand, we provide a lower bound result: we construct a point cloud that requires super-polynomial complexity for a high-quality approximation of the persistence. Together with our approximation schemes, we show that polynomial complexity is achievable for rough approximations, but impossible for sufficiently fine approximations. For some metric spaces, the intrinsic dimension is low in small neighborhoods of the input points, but much higher for large scales of distances. We develop a concept of local intrinsic dimension to capture this property. We also present several applications of this concept, including an approximation method for persistence. This thesis is written in English.Persistent Homology ist eine Methode zur Analyse und Veranschaulichung von Daten aus topologischer Sicht. Sie berechnet eine topologische Zusammenfassung eines metrischen Raumes, die Persistence genannt wird, indem die topologischen Eigenschaften des Raumes über verschiedene Skalen von Abständen analysiert werden. Die Berechnung von Persistence ist für niederdimensionale topologische Eigenschaften effizient. Leider ist die Berechung für mittlere bis hohe Dimensionen sehr teuer. In dieser Dissertation greifen wir dieses Problem aus vielen verschiedenen Winkeln an. Wir stellen effiziente Techniken vor, um die Persistence für metrische Räume zu approximieren. Drei unserer Methoden eignen sich für Punktwolken im euklidischen Raum. Wir verwenden hochdimensionale Gittergeometrie, um die Kosten unserer Approximationen zu reduzieren. Insbesondere entdecken wir mehrere Eigenschaften des Permutahedral Gitters, dessen Voronoi-Zelle für ihre kombinatorischen Eigenschaften bekannt ist. Die vierte Methode eignet sich für Punktwolken mit geringer intrinsischer Dimension: wir verwenden die strukturellen Eigenschaften, um die Komplexität zu reduzieren. Für einige Methoden zeigen wir einen Trade-off zwischen Komplexität und Approximationsqualität auf. Zwei unserer Methoden funktionieren gut mit Dimensionsreduktionstechniken: wir präsentieren die erste Methode mit polynomieller Komplexität unabhängig von der Dimension. Wir zeigen auch eine untere Schranke. Wir konstruieren eine Punktwolke, für die die Berechnung der Persistence nicht in Polynomzeit möglich ist. Die bedeutet, dass in Polynomzeit nur eine grobe Approximation berechnet werden kann. Für gewisse metrische Räume ist die intrinsiche Dimension gering bei kleinen Skalen aber hoch bei großen Skalen. Wir führen das Konzept lokale intrinsische Dimension ein, um diesen Umstand zu fassen, und zeigen, dass es für eine gute Approximation von Persistenz benutzt werden kann. Diese Dissertation ist in englischer Sprache verfasst
On the existence of 0/1 polytopes with high semidefinite extension complexity
In Rothvo\ss{} it was shown that there exists a 0/1 polytope (a polytope
whose vertices are in \{0,1\}^{n}) such that any higher-dimensional polytope
projecting to it must have 2^{\Omega(n)} facets, i.e., its linear extension
complexity is exponential. The question whether there exists a 0/1 polytope
with high PSD extension complexity was left open. We answer this question in
the affirmative by showing that there is a 0/1 polytope such that any
spectrahedron projecting to it must be the intersection of a semidefinite cone
of dimension~2^{\Omega(n)} and an affine space. Our proof relies on a new
technique to rescale semidefinite factorizations
A compact data structure for high dimensional Coxeter-Freudenthal-Kuhn triangulations
We consider a family of highly regular triangulations of Rd that can be stored and queried efficiently in high dimensions. This family consists of Freudenthal-Kuhn triangulations and their images through affine mappings, among which are the celebrated Coxeter triangulations of type Ãd. Those triangulations have major advantages over grids in applications in high dimensions like interpolation of functions and manifold sampling and meshing. We introduce an elegant and very compact data structure to implicitly store the full facial structure of such triangulations. This data structure allows to locate a point and to retrieve the faces or the cofaces of a simplex of any dimension in an output sensitive way. The data structure has been implemented and experimental 9 results are presented
Homological percolation and the Euler characteristic
In this paper we study the connection between the phenomenon of homological
percolation (the formation of "giant" cycles in persistent homology), and the
zeros of the expected Euler characteristic curve. We perform an experimental
study that covers four different models: site-percolation on the cubical and
permutahedral lattices, the Poisson-Boolean model, and Gaussian random fields.
All the models are generated on the flat torus , for . The
simulation results strongly indicate that the zeros of the expected Euler
characteristic curve approximate the critical values for
homological-percolation. Our results also provide some insight about the
approximation error. Further study of this connection could have powerful
implications both in the study of percolation theory, and in the field of
Topological Data Analysis
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
On the existence of 0/1 polytopes with high semidefinite extension complexity
In Rothvoß (Math Program 142(1–2):255–268, 2013) it was shown that
there exists a 0/1 polytope (a polytope whose vertices are in {0, 1}n) such that any
higher-dimensional polytope projecting to it must have 2Ω(n) facets, i.e., its linear
extension complexity is exponential. The question whether there exists a 0/1 polytope
with high positive semidefinite extension complexity was left open. We answer this
question in the affirmative by showing that there is a 0/1 polytope such that any spectrahedron
projecting to it must be the intersection of a semidefinite cone of dimension
2Ω(n) and an affine space. Our proof relies on a new technique to rescale semidefinite
factorizations