9 research outputs found
Towards Stratification Learning through Homology Inference
A topological approach to stratification learning is developed for point
cloud data drawn from a stratified space. Given such data, our objective is to
infer which points belong to the same strata. First we define a multi-scale
notion of a stratified space, giving a stratification for each radius level. We
then use methods derived from kernel and cokernel persistent homology to
cluster the data points into different strata, and we prove a result which
guarantees the correctness of our clustering, given certain topological
conditions; some geometric intuition for these topological conditions is also
provided. Our correctness result is then given a probabilistic flavor: we give
bounds on the minimum number of sample points required to infer, with
probability, which points belong to the same strata. Finally, we give an
explicit algorithm for the clustering, prove its correctness, and apply it to
some simulated data.Comment: 48 page
Data-Driven Analysis of Pareto Set Topology
When and why can evolutionary multi-objective optimization (EMO) algorithms
cover the entire Pareto set? That is a major concern for EMO researchers and
practitioners. A recent theoretical study revealed that (roughly speaking) if
the Pareto set forms a topological simplex (a curved line, a curved triangle, a
curved tetrahedron, etc.), then decomposition-based EMO algorithms can cover
the entire Pareto set. Usually, we cannot know the true Pareto set and have to
estimate its topology by using the population of EMO algorithms during or after
the runtime. This paper presents a data-driven approach to analyze the topology
of the Pareto set. We give a theory of how to recognize the topology of the
Pareto set from data and implement an algorithm to judge whether the true
Pareto set may form a topological simplex or not. Numerical experiments show
that the proposed method correctly recognizes the topology of high-dimensional
Pareto sets within reasonable population size.Comment: 8 pages, accepted at GECCO'18 as a full pape
Triangulating stratified manifolds I: a reach comparison theorem
In this paper, we define the reach for submanifolds of Riemannian manifolds, in a way that is similar to the Euclidean case. Given a d-dimensional submanifold S of a smooth Riemannian manifold M and a point p ∈ M that is not too far from S we want to give bounds on local feature size of exp −1 p (S). Here exp −1 p is the inverse exponential map, a canonical map from the manifold to the tangent space. Bounds on the local feature size of exp −1 p (S) can be reduced to giving bounds on the reach of exp −1 p (B), where B is a geodesic ball, centred at c with radius equal to the reach of S. Equivalently we can give bounds on the reach of exp −1 p • exp c (B c), where now B c is a ball in the tangent space T c M, with the same radius. To establish bounds on the reach of exp −1 p • exp c (B c) we use bounds on the metric and on its derivative in Riemann normal coordinates. This result is a first step towards answering the important question of how to triangulate stratified manifolds
The topological correctness of PL approximations of isomanifolds
Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f : Rd → Rd−n. A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation T of the ambient space Rd. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently
fine triangulation T . This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary