Shape-Preserving Dimensionality Reduction : An Algorithm and Measures of Topological Equivalence

We introduce a linear dimensionality reduction technique preserving topological features via persistent homology. The method is designed to find linear projection \(L\) which preserves the persistent diagram of a point cloud \(\mathbb{X}\) via simulated annealing. The projection \(L\) induces a set...

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Bibliographic Details
Published in:arXiv.org 2021-06
Main Authors: Yu, Byeongsu, You, Kisung
Format: Article
Language:English
Subjects:
Algorithms
Equivalence
Filtration
Homology
Homomorphisms
Isomorphism
Projection
Reduction
Simulated annealing
Topology
Online Access:Get full text
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Summary:We introduce a linear dimensionality reduction technique preserving topological features via persistent homology. The method is designed to find linear projection \(L\) which preserves the persistent diagram of a point cloud \(\mathbb{X}\) via simulated annealing. The projection \(L\) induces a set of canonical simplicial maps from the Rips (or Čech) filtration of \(\mathbb{X}\) to that of \(L\mathbb{X}\). In addition to the distance between persistent diagrams, the projection induces a map between filtrations, called filtration homomorphism. Using the filtration homomorphism, one can measure the difference between shapes of two filtrations directly comparing simplicial complexes with respect to quasi-isomorphism \(\mu_{\operatorname{quasi-iso}}\) or strong homotopy equivalence \(\mu_{\operatorname{equiv}}\). These \(\mu_{\operatorname{quasi-iso}}\) and \(\mu_{\operatorname{equiv}}\) measures how much portion of corresponding simplicial complexes is quasi-isomorphic or homotopy equivalence respectively. We validate the effectiveness of our framework with simple examples.
ISSN:2331-8422