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Abstract

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.