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- W2769896813 abstract "Classical methods to model topological properties of point clouds, such as the Vietoris–Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration of the Rips complex with improved bounds for size: precisely, for n points in $$mathbb {R}^d$$ , we obtain a O(d)-approximation whose k-skeleton has size $$n2^{O(d log k)}$$ per scale and $$n2^{O(dlog d)}$$ in total over all scales. In conjunction with dimension reduction techniques, our approach yields a $$O(mathrm {polylog} (n))$$ -approximation of size $$n^{O(1)}$$ for Rips filtrations on arbitrary metric spaces. This result stems from high-dimensional lattice geometry and exploits properties of the permutahedral lattice, a well-studied structure in discrete geometry. Building on the same geometric concept, we also present a lower bound result on the size of an approximation: we construct a point set for which every $$(1+varepsilon )$$ -approximation of the Cech filtration has to contain $$n^{Omega (log log n)}$$ features, provided that $$varepsilon <frac{1}{log ^{1+c} n}$$ for $$cin (0,1)$$ ." @default.
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- W2769896813 date "2017-11-13" @default.
- W2769896813 modified "2023-09-29" @default.
- W2769896813 title "Polynomial-Sized Topological Approximations Using the Permutahedron" @default.
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- W2769896813 doi "https://doi.org/10.1007/s00454-017-9951-2" @default.
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