Shape-Preserving Dimensionality Reduction : An Algorithm and Measures of Topological Equivalence
This work addresses the challenge of maintaining shape integrity in data analysis for researchers in computational topology, though it appears incremental as it builds on existing persistent homology methods.
The paper tackles the problem of dimensionality reduction while preserving topological features, introducing a linear technique that uses persistent homology and simulated annealing to find projections that maintain the persistent diagram of point clouds, with validation on simple examples.
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 $μ_{\operatorname{quasi-iso}}$ or strong homotopy equivalence $μ_{\operatorname{equiv}}$. These $μ_{\operatorname{quasi-iso}}$ and $μ_{\operatorname{equiv}}$ 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.