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Topological data analysis combines machine learning with methods from algebraic topology. Persistent homology, a method to characterize topological features occurring in data at multiple scales is of particular interest. A major obstacle to the ...

We show that the nerve and clique complexes of n arcs in the circle are homotopy equivalent to either a point, an odd-dimensional sphere, or a wedge sum of spheres of the same even dimension. Moreover this homotopy type can be computed in time $$O(n\log ...

Given a set of points that sample a shape, the Rips complex of the points is often used to provide an approximation of the shape easily-computed. It has been proved that the Rips complex captures the homotopy type of the shape, assuming that the ...

We study the homology of simplicial complexes built via deterministic rules from a random set of vertices. In particular, we show that, depending on the randomness that generates the vertices, the homology of these complexes can either become trivial as ...

In many applications, the first step into the topological analysis of a discrete point set P sampled from a manifold is the construction of a simplicial complex with vertices on P. In this paper, we present an algorithm for the efficient computation of ...

The flow complex is a geometric structure, similar to the Delaunay tessellation, to organize a set of (weighted) points in R^k. Flow shapes are topological spaces corresponding to substructures of the flow complex. The flow complex and flow shapes have ...