Clustering with UMAP: Why and How Connectivity Matters

Abstract

Topology based dimensionality reduction methods such as t-SNE and UMAP have seen increasing success and popularity in highdimensional data. These methods have strong mathematical foundations and are based on the intuition that the topology in low dimensions should be close to that of high dimensions. Given that the initial topological structure is a precursor to the success of the algorithm, this naturally raises the question: What makes a "good" topological structure for dimensionality reduction? In this paper which focuses on UMAP, we study the effects of node connectivity (k-Nearest Neighbors vs mutual k-Nearest Neighbors) and relative neighborhood (Adjacent via Path Neighbors) on dimensionality reduction. We explore these concepts through extensive ablation studies on 4 standard image and text datasets; MNIST, FMNIST, 20NG, AG, reducing to 2 and 64 dimensions. Our findings indicate that a more refined notion of connectivity (mutual k-Nearest Neighbors with minimum spanning tree) together with a flexible method of constructing the local neighborhood (Path Neighbors), can achieve a much better representation than default UMAP, as measured by downstream clustering performance.

Molecularly stratified hypothalamic astrocytes are cellular foci for obesity

“…Secondly, the UMAP algorithm drove the 10-NN graph from this matrix in Euclidean space with pynndescent optimization. Thirdly, a 20-NN graph and distance matrix was used to build a 20-mutual-NN graph using the minimal tree path-connectivity model (functions: min_spanning_tree(), create_connected_graph(), nd_new_nn(), mutual_nn_nearest() in microclusters_to_groups_cv_feature_selection_svc.ipynb) 127 . Fourthly, we derived a fuzzy simplical set for the ensuing graph using the UMAP algorithm 136 .…”