2024/02/23 by Wagner, Hubert, Arustamyan, Nickolas, Wheeler, Matthew +1 · 3 citations
#Algebraic Topology (math.AT) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG)
paper · doi:10.48550/arxiv.2402.15058
We combine standard persistent homology with image persistent homology to define a novel way of characterizing shapes and interactions between them. In particular, we introduce: (1) a mixup barcode, which captures geometric-topological interactions (mixup) between two point sets in arbitrary dimension; (2) simple summary statistics, total mixup and total percentage mixup, which quantify the complexity of the interactions as a single number; (3) a software tool for playing with the above. As a proof of concept, we apply this tool to a problem arising from machine learning. In particular, we study the disentanglement in embeddings of different classes. The results suggest that topological mixup is a useful method for characterizing interactions for low and high-dimensional data. Compared to the typical usage of persistent homology, the new tool is sensitive to the geometric locations of the topological features, which is often desirable.