Related papers: Multiscale 2-Mapper -- Exploratory Data Analysis G…

Mapper algorithm can be used to build graph-based representations of high-dimensional data capturing structurally interesting features such as loops, flares or clusters. The graph can be further annotated with additional colouring of…

Topological data analysis offers a robust way to extract useful information from noisy, unstructured data by identifying its underlying structure. Recently, an efficient quantum algorithm was proposed [Lloyd, Garnerone, Zanardi, Nat.…

Data analysis in high-dimensional spaces aims at obtaining a synthetic description of a data set, revealing its main structure and its salient features. We here introduce an approach providing this description in the form of a topography of…

Acquiring plausible pathways on high-dimensional structural distributions is beneficial in several domains. For example, in the drug discovery field, a protein conformational pathway, i.e. a highly probable sequence of protein structural…

Cumulant mapping employs a statistical reconstruction of the whole by sampling its parts. The theory developed in this work formalises and extends ad hoc methods of `multi-fold' or `multi-dimensional' covariance mapping. Explicit formulae…

The empirical results suggest that the learnability of a neural network is directly related to its size. To mathematically prove this, we borrow a tool in topological algebra: Betti numbers to measure the topological geometric complexity of…

The Mapper construction is one of the most widespread tools from Topological Data Analysis. There is an unfortunate trend as the construction has gained traction to use clustering methods with properties that end up distorting any analysis…

We specialize techniques from topological data analysis to the problem of characterizing the topological complexity (as defined in the body of the paper) of a multi-class data set. As a by-product, a topological classifier is defined that…

Multi-objective optimization is a crucial matter in computer systems design space exploration because real-world applications often rely on a trade-off between several objectives. Derivatives are usually not available or impractical to…

The predictions of mean-field electrodynamics can now be probed using direct numerical simulations of random flows and magnetic fields. When modelling astrophysical MHD, it is important to verify that such simulations are in agreement with…

The Mapper algorithm is a popular tool for visualization and data exploration in topological data analysis. We investigate an inverse problem for the Mapper algorithm: Given a dataset $X$ and a graph $G$, does there exist a set of Mapper…

This paper introduces the concept of a bi-scale metric for use in the cooperative phase of the self-organizing map (SOM) algorithm. Use of a bi-scale metric allows segmentation of the map into a number of regions, corresponding to…

High-throughput technologies to collect field data have made observations possible at scale in several branches of life sciences. The data collected can range from the molecular level (genotypes) to physiological (phenotypic traits) and…

Persistent homology has emerged as a popular technique for the topological simplification of big data, including biomolecular data. Multidimensional persistence bears considerable promise to bridge the gap between geometry and topology.…

Computational analyses of, e.g., genomic, proteomic, or metabolomic data, commonly result in one or more sets of candidate genes, proteins, or enzymes. These sets are often the outcome of clustering algorithms. Subsequently, it has to be…

The mapper construction is a powerful tool from topological data analysis that is designed for the analysis and visualization of multivariate data. In this paper, we investigate a method for stitching a pair of univariate mappers together…

We study the topological construction called Mapper in the context of simply connected domains, in particular on images. The Mapper construction can be considered as a generalization for contour, split, and joint trees on simply connected…

Clustering aims to form groups of similar data points in an unsupervised regime. Yet, clustering complex datasets containing critically intertwined shapes poses significant challenges. The prevailing clustering algorithms widely depend on…

Topological data analysis is an emerging field that applies the study of topological invariants to data. Perhaps the simplest of these invariants is the number of connected components or clusters. In this work, we explore a topological…

Large-scale vector mapping is important for transportation, city planning, and survey and census. We propose GraphMapper, a unified framework for end-to-end vector map extraction from satellite images. Our key idea is a novel unified…