Mapper and Ball Mapper are Topological Data Analysis tools used for exploring high dimensional point clouds and visualizing scalar–valued functions on those point clouds. Inspired by open questions in knot theory, new features are added to Ball Mapper that enable encoding of the structure, internal relations and symmetries of the point cloud. Moreover, the strengths of Mapper and Ball Mapper constructions are combined to create a tool for comparing high dimensional data descriptors of a single dataset. This new hybrid algorithm, Mapper on Ball Mapper, is applicable to high dimensional lens functions. As a proof of concept we include applications to knot and game theory.
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Mapper Interactive: A Scalable, Extendable, and Interactive Toolbox for the Visual Exploration of High-Dimensional Data
The mapper algorithm is a popular tool from topological data analysis for extracting topological summaries of high-dimensional datasets. In this paper, we present Mapper Interactive, a web-based framework for the interactive analysis and visualization of high-dimensional point cloud data. It implements the mapper algorithm in an interactive, scalable, and easily extendable way, thus supporting practical data analysis. In particular, its command-line API can compute mapper graphs for 1 million points of 256 dimensions in about 3 minutes (4 times faster than the vanilla implementation). Its visual interface allows on-the-fly computation and manipulation of the mapper graph based on user-specified parameters and supports the addition of new analysis modules with a few lines of code. Mapper Interactive makes the mapper algorithm accessible to nonspecialists and accelerates topological analytics workflows.
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- Award ID(s):
- 1661375
- NSF-PAR ID:
- 10310710
- Date Published:
- Journal Name:
- IEEE 14th Pacific Visualization Symposium (PacificVis)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Simplicial complexes are widely used to discretize shapes. In low dimensions, a 3D shape is represented by discretizing its boundary surface, encoded as a triangle mesh, or by discretizing the enclosed volume, encoded as a tetrahedral mesh. High‐dimensional simplicial complexes have recently found their application in topological data analysis. Topological data analysis aims at studying a point cloud P, possibly embedded in a high‐dimensional metric space, by investigating the topological characteristics of the simplicial complexes built on P. Analysing such complexes is not feasible due to their size and dimensions. To this aim, the idea of simplifying a complex while preserving its topological features has been proposed in the literature. Here, we consider the problem of efficiently simplifying simplicial complexes in arbitrary dimensions. We provide a new definition for the edge contraction operator, based on a top‐based data structure, with the objective of preserving structural aspects of a simplicial shape (i.e., its homology), and a new algorithm for verifying the link condition on a top‐based representation. We implement the simplification algorithm obtained by coupling the new edge contraction and the link condition on a specific top‐based data structure, that we use to demonstrate the scalability of our approach.
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Deep models are known to be vulnerable to data adversarial attacks, and many adversarial training techniques have been developed to improve their adversarial robustness. While data adversaries attack model predictions through modifying data, little is known about their impact on the neuron activations produced by the model, which play a crucial role in determining the model’s predictions and interpretability. In this work, we aim to develop a topological understanding of adversarial training to enhance its interpretability. We analyze the topological structure—in particular, mapper graphs—of neuron activations of data samples produced by deep adversarial training. Each node of a mapper graph represents a cluster of activations, and two nodes are connected by an edge if their corresponding clusters have a nonempty intersection. We provide an interactive visualization tool that demonstrates the utility of our topological framework in exploring the activation space. We found that stronger attacks make the data samples more indistinguishable in the neuron activation space that leads to a lower accuracy. Our tool also provides a natural way to identify the vulnerable data samples that may be useful in improving model robustness.more » « less
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This paper introduces a web-based interactive educational platform for 3D/polyhedral graphic statics (PGS) [1]. The Block Research Group (BRG) at ETH Zürich developed a dynamic learning and teaching platform for structural design. This tool is based on traditional graphic statics. It uses interactive 2D drawings to help designers and engineers with all skill levels to understand and utilize the methods [2]. However, polyhedral graphic statics is not easy to learn because of its characteristics in three-dimensional. All the existing computational design tools are heavily dependent on the modeling software such as Rhino or the Python-based computational framework like Compass [3]. In this research, we start with the procedural approach, developing libraries using JavaScript, Three.js, and WebGL to facilitate the construction by making it independent from any software. This framework is developed based on the mathematical and computational algorithms deriving the global equilibrium of the structure, optimizing the balanced relationship between the external magnitudes and the internal forces, visualizing the dynamic reciprocal polyhedral diagrams with corresponding topological data. This instant open-source application and the visualization interface provide a more operative platform for students, educators, practicers, and designers in an interactive manner, allowing them to learn not only the topological relationship but also to deepen their knowledge and understanding of structures in the steps for the construction of the form and force diagrams and analyze it. In the simplified single-node example, the multi-step geometric procedures intuitively illustrate 3D structural reciprocity concepts. With the intuitive control panel, the user can move the constraint point’s location through the inserted gumball function, the force direction of the form diagram will be dynamically changed from compression-only to tension and compression combined. Users can also explore and design innovative, efficient spatial structures with changeable boundary conditions and constraints through real-time manipulating both force distribution and geometric form, such as adding the number of supports or subdividing the global equilibrium in the force diagram. Eventually, there is an option to export the satisfying geometry as a suitable format to share with other fabrication tools. As the online educational environment with different types of geometric examples, it is valuable to use graphical approaches to teach the structural form in an exploratory manner.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/PacificVis52677.2021.00021
