Topology-Preserving Dimensionality Reduction via Interleaving Optimization
This addresses the need for reliable topological preservation in embeddings for data preprocessing and visualization, though it appears incremental as it builds on existing interleaving distance concepts.
The paper tackles the problem of ensuring topological correctness in dimensionality reduction by minimizing the interleaving distance between persistent homology filtrations, and demonstrates its application to find optimal linear projections for data visualization.
Dimensionality reduction techniques are powerful tools for data preprocessing and visualization which typically come with few guarantees concerning the topological correctness of an embedding. The interleaving distance between the persistent homology of Vietoris-Rips filtrations can be used to identify a scale at which topological features such as clusters or holes in an embedding and original data set are in correspondence. We show how optimization seeking to minimize the interleaving distance can be incorporated into dimensionality reduction algorithms, and explicitly demonstrate its use in finding an optimal linear projection. We demonstrate the utility of this framework to data visualization.