Manifold Learning via Data Topology Optimization: Gradient Method Application

Julia Borisova, Alexander Hvatov

Keywords: manifold learning, gradient optimization, Isomap, topology search

TL;DR: Paper propose approach for intrinsic data topolody search with end-to-end differentiable Isomap implementation and obtained manifold usage for out-of sample mapping.

Abstract: 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 for inferring the underlying topological structure through joint optimization of a learnable distance matrix and embedding. Our approach combines the learning of neural networks with a differentiable Isomap implementation, enabling 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.

Primary Area: learning on graphs and other geometries & topologies

Submission Number: 3439