GK-Mapper: A Stability Framework for Gustafson-Kessel Fuzzy Mapper Graphs
Topological Data Analysis uses tools from algebraic topology to study the shape and structure of data. The Mapper algorithm provides a graph-based summary of high-dimensional datasets by combining a filter function, a cover of the filter range, and clustering on the corresponding pullback sets. Several variants of Mapper have been proposed, including Conventional Mapper, F-Mapper, and Shape Fuzzy C-Means Mapper. In this article, we introduce Gustafson-Kessel Fuzzy Mapper Graphs, a geometry-adaptive extension of Shape Fuzzy C-Means Mapper. The proposed method replaces spherical fuzzy covers with ellipsoidal covers induced by the Gustafson-Kessel fuzzy clustering framework, making it more suitable for high-dimensional datasets with anisotropic and non-spherical geometry. We develop a stability framework for the graphs produced by Gustafson-Kessel Mapper and Shape Fuzzy C-Means Mapper. We prove that the membership functions depend smoothly on the fuzzifier, establish a precise condition for the existence of edges, and show that the graph is locally stable under small perturbations of the fuzzifier. We further describe the critical-event structure of graph changes in terms of threshold crossings of the membership functions and show that the graph is constant between consecutive critical events. When the threshold-crossing set is finite, this yields an eventual freezing threshold. Finally, we empirically show that Gustafson-Kessel Mapper can produce more stable graphs than Shape Fuzzy C-Means Mapper on high-dimensional and geometrically complex datasets.