A Curvature-Based Adaptive UMAP for Manifold Learning in Clustering and Classification

• Gustavo Henrique Chavari UFSCar
• Alexandre Luís Magalhães Levada UFSCar

Resumo

The Uniform Manifold Approximation and Projection (UMAP) algorithm is a powerful tool for dimensionality reduction, renowned for its ability to preserve both local and global data structures. However, its efficacy relies on a critical hyperparameter: the number of neighbors, which remains fixed for all data points. This rigidity can be suboptimal, as the intrinsic dimensionality and density can vary significantly across a manifold. In this paper, we introduce SH-UMAP, an extension that determines the neighborhood size for each point adaptively. SH-UMAP estimates the local curvature of the data manifold using a numerical approximation of the shape operator. Points in high-curvature regions are assigned smaller neighborhoods, while points in flatter regions are given larger ones. We present a preliminary experimental evaluation on benchmark datasets. The results indicate that the embeddings generated by SH-UMAP lead to improved performance in both unsupervised clustering and supervised classification tasks when compared to the standard UMAP algorithm, demonstrating the practical benefits of an adaptive-neighborhood approach.

Palavras-chave: Manifolds, Shape, Clustering algorithms, Benchmark testing, Data structures, Robustness, Manifold learning, Classification algorithms, Rigidity, Standards