Mixup Barcodes: Quantifying Geometric-Topological Interactions between Point Clouds
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.
Code (0)
등록된 구현이 없습니다.
Tasks
DisentanglementMethods 이 논문이 사용한 방법론
Similar Papers 제목 키워드 기반
Comparing the Effects of Persistence Barcodes Aggregation and Feature Concatenation on Medical Imaging
In medical image analysis, feature engineering plays an important role in the design and performance of machine learning models. Persistent homology (PH), from the field of topological data analysis (TDA), demonstrates r…
Feature EngineeringMedical Image AnalysisTopological Data AnalysisPersistent Homology via Ellipsoids
Persistent homology is one of the most popular methods in Topological Data Analysis. An initial step in any analysis with persistent homology involves constructing a nested sequence of simplicial complexes, called a filt…
Topological Data AnalysisTopological Classification in a Wasserstein Distance Based Vector Space
Classification of large and dense networks based on topology is very difficult due to the computational challenges of extracting meaningful topological features from real-world networks. In this paper we present a comput…
ClassificationDuality in Persistent Homology of Images
We derive the relationship between the persistent homology barcodes of two dual filtered CW complexes. Applied to greyscale digital images, we obtain an algorithm to convert barcodes between the two different (dual) topo…
Barcodes as Summary of Loss Function Topology
We propose to study neural networks' loss surfaces by methods of topological data analysis. We suggest to apply barcodes of Morse complexes to explore topology of loss surfaces. An algorithm for calculations of the loss …
Topological Data Analysis