350 research outputs found
Categorification of persistent homology
We redevelop persistent homology (topological persistence) from a categorical
point of view. The main objects of study are diagrams, indexed by the poset of
real numbers, in some target category. The set of such diagrams has an
interleaving distance, which we show generalizes the previously-studied
bottleneck distance. To illustrate the utility of this approach, we greatly
generalize previous stability results for persistence, extended persistence,
and kernel, image and cokernel persistence. We give a natural construction of a
category of interleavings of these diagrams, and show that if the target
category is abelian, so is this category of interleavings.Comment: 27 pages, v3: minor changes, to appear in Discrete & Computational
Geometr
Optimal topological simplification of discrete functions on surfaces
We solve the problem of minimizing the number of critical points among all
functions on a surface within a prescribed distance {\delta} from a given input
function. The result is achieved by establishing a connection between discrete
Morse theory and persistent homology. Our method completely removes homological
noise with persistence less than 2{\delta}, constructively proving the
tightness of a lower bound on the number of critical points given by the
stability theorem of persistent homology in dimension two for any input
function. We also show that an optimal solution can be computed in linear time
after persistence pairs have been computed.Comment: 27 pages, 8 figure
Computational Topology Techniques for Characterizing Time-Series Data
Topological data analysis (TDA), while abstract, allows a characterization of
time-series data obtained from nonlinear and complex dynamical systems. Though
it is surprising that such an abstract measure of structure - counting pieces
and holes - could be useful for real-world data, TDA lets us compare different
systems, and even do membership testing or change-point detection. However, TDA
is computationally expensive and involves a number of free parameters. This
complexity can be obviated by coarse-graining, using a construct called the
witness complex. The parametric dependence gives rise to the concept of
persistent homology: how shape changes with scale. Its results allow us to
distinguish time-series data from different systems - e.g., the same note
played on different musical instruments.Comment: 12 pages, 6 Figures, 1 Table, The Sixteenth International Symposium
on Intelligent Data Analysis (IDA 2017
Computation of protein geometry and its applications: Packing and function prediction
This chapter discusses geometric models of biomolecules and geometric
constructs, including the union of ball model, the weigthed Voronoi diagram,
the weighted Delaunay triangulation, and the alpha shapes. These geometric
constructs enable fast and analytical computaton of shapes of biomoleculres
(including features such as voids and pockets) and metric properties (such as
area and volume). The algorithms of Delaunay triangulation, computation of
voids and pockets, as well volume/area computation are also described. In
addition, applications in packing analysis of protein structures and protein
function prediction are also discussed.Comment: 32 pages, 9 figure
Dualities in persistent (co)homology
We consider sequences of absolute and relative homology and cohomology groups
that arise naturally for a filtered cell complex. We establish algebraic
relationships between their persistence modules, and show that they contain
equivalent information. We explain how one can use the existing algorithm for
persistent homology to process any of the four modules, and relate it to a
recently introduced persistent cohomology algorithm. We present experimental
evidence for the practical efficiency of the latter algorithm.Comment: 16 pages, 3 figures, submitted to the Inverse Problems special issue
on Topological Data Analysi
Persistence modules, shape description, and completeness
Persistence modules are algebraic constructs that can be used to describe the shape of an object starting from a geometric representation of it. As shape descriptors, persistence modules are not complete, that is they may not distinguish non-equivalent shapes.
In this paper we show that one reason for this is that homomorphisms between persistence modules forget the geometric nature of the problem. Therefore we introduce geometric homomorphisms between persistence modules, and show that in some cases they perform better. A combinatorial structure, the -tree, is shown to be an invariant for geometric isomorphism classes in the case of persistence modules obtained through the 0th persistent homology functor
Searching edges in the overlap of two plane graphs
Consider a pair of plane straight-line graphs, whose edges are colored red
and blue, respectively, and let n be the total complexity of both graphs. We
present a O(n log n)-time O(n)-space technique to preprocess such pair of
graphs, that enables efficient searches among the red-blue intersections along
edges of one of the graphs. Our technique has a number of applications to
geometric problems. This includes: (1) a solution to the batched red-blue
search problem [Dehne et al. 2006] in O(n log n) queries to the oracle; (2) an
algorithm to compute the maximum vertical distance between a pair of 3D
polyhedral terrains one of which is convex in O(n log n) time, where n is the
total complexity of both terrains; (3) an algorithm to construct the Hausdorff
Voronoi diagram of a family of point clusters in the plane in O((n+m) log^3 n)
time and O(n+m) space, where n is the total number of points in all clusters
and m is the number of crossings between all clusters; (4) an algorithm to
construct the farthest-color Voronoi diagram of the corners of n axis-aligned
rectangles in O(n log^2 n) time; (5) an algorithm to solve the stabbing circle
problem for n parallel line segments in the plane in optimal O(n log n) time.
All these results are new or improve on the best known algorithms.Comment: 22 pages, 6 figure
Algorithms for Colourful Simplicial Depth and Medians in the Plane
The colourful simplicial depth of a point x in the plane relative to a
configuration of n points in k colour classes is exactly the number of closed
simplices (triangles) with vertices from 3 different colour classes that
contain x in their convex hull. We consider the problems of efficiently
computing the colourful simplicial depth of a point x, and of finding a point,
called a median, that maximizes colourful simplicial depth.
For computing the colourful simplicial depth of x, our algorithm runs in time
O(n log(n) + k n) in general, and O(kn) if the points are sorted around x. For
finding the colourful median, we get a time of O(n^4). For comparison, the
running times of the best known algorithm for the monochrome version of these
problems are O(n log(n)) in general, improving to O(n) if the points are sorted
around x for monochrome depth, and O(n^4) for finding a monochrome median.Comment: 17 pages, 8 figure
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