10,174 research outputs found
The Depth Poset of a Filtered Lefschetz Complex
Taking a discrete approach to functions and dynamical systems, this paper
integrates the combinatorial gradients in Forman's discrete Morse theory with
persistent homology to forge a unified approach to function simplification. The
two crucial ingredients in this effort are the Lefschetz complex, which focuses
on the homology at the expense of the geometry of the cells, and the shallow
pairs, which are birth-death pairs that can double as vectors in discrete Morse
theory. The main new concept is the depth poset on the birth-death pairs, which
captures all simplifications achieved through canceling shallow pairs. One of
its linear extensions is the ordering by persistence
\v{C}ech-Delaunay gradient flow and homology inference for self-maps
We call a continuous self-map that reveals itself through a discrete set of
point-value pairs a sampled dynamical system. Capturing the available
information with chain maps on Delaunay complexes, we use persistent homology
to quantify the evidence of recurrent behavior. We establish a sampling theorem
to recover the eigenspace of the endomorphism on homology induced by the
self-map. Using a combinatorial gradient flow arising from the discrete Morse
theory for \v{C}ech and Delaunay complexes, we construct a chain map to
transform the problem from the natural but expensive \v{C}ech complexes to the
computationally efficient Delaunay triangulations. The fast chain map algorithm
has applications beyond dynamical systems.Comment: 22 pages, 8 figure
Euler Integration of Gaussian Random Fields and Persistent Homology
In this paper we extend the notion of the Euler characteristic to persistent
homology and give the relationship between the Euler integral of a function and
the Euler characteristic of the function's persistent homology. We then proceed
to compute the expected Euler integral of a Gaussian random field using the
Gaussian kinematic formula and obtain a simple closed form expression. This
results in the first explicitly computable mean of a quantitative descriptor
for the persistent homology of a Gaussian random field.Comment: 21 pages, 1 figur
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
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