Dimensionality Reduction for k-Distance Applied to Persistent Homology

@article{Arya2020DimensionalityRF,
  title={Dimensionality Reduction for k-Distance Applied to Persistent Homology},
  author={Shreya Arya and Jean-Daniel Boissonnat and Kunal Dutta and Martin Lotz},
  journal={ArXiv},
  year={2020},
  volume={abs/2110.05897}
}
Given a set P of n points and a constant k, we are interested in computing the persistent homology of the Cech filtration of P for the k-distance, and investigate the effectiveness of dimensionality reduction for this problem, answering an open question of Sheehy [Proc. SoCG, 2014]. We first show using the Johnson-Lindenstrauss lemma, that the persistent homology can be preserved up to a (1 ± e) factor while reducing dimensionality to O(k log n/e2). Our main result shows that the target… 
Shape-Preserving Dimensionality Reduction : An Algorithm and Measures of Topological Equivalence
TLDR
A linear dimensionality reduction technique preserving topological features via persistent homology, designed to find linear projection L which preserves the persistent diagram of a point cloud X via simulated annealing, is introduced.

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