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

• URL: http://arxiv.org/abs/2106.02096v2
• Date: Mon, 7 Jun 2021 16:59:07 GMT
• Title: Shape-Preserving Dimensionality Reduction : An Algorithm and Measures of Topological Equivalence
• Authors: Byeongsu Yu, Kisung You
• Abstract 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 $mathbbX$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: 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 \v{C}ech) 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.

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