Towards Intrinsic Topology Search: Differentiable Isomap Application

The study of topological properties in data and their application to machine learning is a growing research area. While most methods operate in Euclidean space, alternative topologies (e.g., hyperbolic embeddings for recommender systems) often yield superior performance. However, real-world data sets lack a known intrinsic topology, which requires manual specification. We propose a novel method to infer the underlying topological structure through the joint optimization of a learnable distance matrix and embedding. Our approach combines the learning of neural networks with a differentiable Isomap implementation, enabling the end-to-end optimization of both the metric and mapping. Experiments on synthetic non-Euclidean datasets demonstrate accurate topology recovery, suggesting broader applicability to real-world problems with unknown geometric structure, a claim we preliminarily validate on the MNIST dataset.