🍩 Database of Original & Non-Theoretical Uses of Topology

(found 2 matches in 0.000904s)
  1. Generalized Penalty for Circular Coordinate Representation (2020)

    Hengrui Luo, Alice Patania, Jisu Kim, Mikael Vejdemo-Johansson
    Abstract Topological Data Analysis (TDA) provides novel approaches that allow us to analyze the geometrical shapes and topological structures of a dataset. As one important application, TDA can be used for data visualization and dimension reduction. We follow the framework of circular coordinate representation, which allows us to perform dimension reduction and visualization for high-dimensional datasets on a torus using persistent cohomology. In this paper, we propose a method to adapt the circular coordinate framework to take into account sparsity in high-dimensional applications. We use a generalized penalty function instead of an \$L_\2\\$ penalty in the traditional circular coordinate algorithm. We provide simulation experiments and real data analysis to support our claim that circular coordinates with generalized penalty will accommodate the sparsity in high-dimensional datasets under different sampling schemes while preserving the topological structures.
  2. Fibers of Failure: Classifying Errors in Predictive Processes (2020)

    Leo S. Carlsson, Mikael Vejdemo-Johansson, Gunnar Carlsson, Pär G. Jönsson
    Abstract Predictive models are used in many different fields of science and engineering and are always prone to make faulty predictions. These faulty predictions can be more or less malignant depending on the model application. We describe fibers of failure (FiFa), a method to classify failure modes of predictive processes. Our method uses Mapper, an algorithm from topological data analysis (TDA), to build a graphical model of input data stratified by prediction errors. We demonstrate two ways to use the failure mode groupings: either to produce a correction layer that adjusts predictions by similarity to the failure modes; or to inspect members of the failure modes to illustrate and investigate what characterizes each failure mode. We demonstrate FiFa on two scenarios: a convolutional neural network (CNN) predicting MNIST images with added noise, and an artificial neural network (ANN) predicting the electrical energy consumption of an electric arc furnace (EAF). The correction layer on the CNN model improved its prediction accuracy significantly while the inspection of failure modes for the EAF model provided guiding insights into the domain-specific reasons behind several high-error regions.