565 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
Lattice Gas Automata for Reactive Systems
Reactive lattice gas automata provide a microscopic approachto the dynamics
of spatially-distributed reacting systems. After introducing the subject within
the wider framework of lattice gas automata (LGA) as a microscopic approach to
the phenomenology of macroscopic systems, we describe the reactive LGA in terms
of a simple physical picture to show how an automaton can be constructed to
capture the essentials of a reactive molecular dynamics scheme. The statistical
mechanical theory of the automaton is then developed for diffusive transport
and for reactive processes, and a general algorithm is presented for reactive
LGA. The method is illustrated by considering applications to bistable and
excitable media, oscillatory behavior in reactive systems, chemical chaos and
pattern formation triggered by Turing bifurcations. The reactive lattice gas
scheme is contrasted with related cellular automaton methods and the paper
concludes with a discussion of future perspectives.Comment: to appear in PHYSICS REPORTS, 81 revtex pages; uuencoded gziped
postscript file; figures available from [email protected] or
[email protected]
Ground States of Fermionic lattice Hamiltonians with Permutation Symmetry
We study the ground states of lattice Hamiltonians that are invariant under
permutations, in the limit where the number of lattice sites, N -> \infty. For
spin systems, these are product states, a fact that follows directly from the
quantum de Finetti theorem. For fermionic systems, however, the problem is very
different, since mode operators acting on different sites do not commute, but
anti-commute. We construct a family of fermionic states, \cal{F}, from which
such ground states can be easily computed. They are characterized by few
parameters whose number only depends on M, the number of modes per lattice
site. We also give an explicit construction for M=1,2. In the first case,
\cal{F} is contained in the set of Gaussian states, whereas in the second it is
not. Inspired by that constructions, we build a set of fermionic variational
wave functions, and apply it to the Fermi-Hubbard model in two spatial
dimensions, obtaining results that go beyond the generalized Hartree-Fock
theory.Comment: 23 pages, published versio
Approximation algorithms for Vietoris-Rips and Äech 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 ĂŒber verschiedene Skalen von AbstĂ€nden analysiert werden. Die Berechnung von Persistence ist fĂŒr niederdimensionale topologische Eigenschaften effizient. Leider ist die Berechung fĂŒr 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 fĂŒr metrische RĂ€ume zu approximieren. Drei unserer Methoden eignen sich fĂŒr 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 fĂŒr ihre kombinatorischen Eigenschaften bekannt ist. Die vierte Methode eignet sich fĂŒr Punktwolken mit geringer intrinsischer Dimension: wir verwenden die strukturellen Eigenschaften, um die KomplexitĂ€t zu reduzieren. FĂŒr einige Methoden zeigen wir einen Trade-off zwischen KomplexitĂ€t und ApproximationsqualitĂ€t auf. Zwei unserer Methoden funktionieren gut mit Dimensionsreduktionstechniken: wir prĂ€sentieren die erste Methode mit polynomieller KomplexitĂ€t unabhĂ€ngig von der Dimension. Wir zeigen auch eine untere Schranke. Wir konstruieren eine Punktwolke, fĂŒr die die Berechnung der Persistence nicht in Polynomzeit möglich ist. Die bedeutet, dass in Polynomzeit nur eine grobe Approximation berechnet werden kann. FĂŒr gewisse metrische RĂ€ume ist die intrinsiche Dimension gering bei kleinen Skalen aber hoch bei groĂen Skalen. Wir fĂŒhren das Konzept lokale intrinsische Dimension ein, um diesen Umstand zu fassen, und zeigen, dass es fĂŒr eine gute Approximation von Persistenz benutzt werden kann. Diese Dissertation ist in englischer Sprache verfasst
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