TADD: A Computational Framework for Data Analysis Using Discrete Morse Theory

Abstract

Abstract. This paper presents a computational framework that allows for a robust extraction of the extremal structure of scalar and vector fields on 2D manifolds embedded in 3D. This structure consists of critical points, separatrices, and periodic orbits. The framework is based on Forman's discrete Morse theory, which guarantees the topological consistency of the computed extremal structure. Using a graph theoretical formulation of this theory, we present an algorithmic pipeline that computes a hierarchy of extremal structures. This hierarchy is defined by an importance measure and enables the user to select an appropriate level of detail. Keywords: Discrete Morse theory, data analysis, scalar fields, vector fields MotivationWe propose a computational framework to extract the extremal structure of scalar and vector fields on 2D manifolds embedded in R 3 . The extremal structure of a scalar field consists of critical points and separatrices -the streamlines of the gradient field that connect the critical points. The extremal structure of a vector field additionally includes periodic orbits -the streamlines that are closed.These structures are of great interest in many applications and have a long history [2,12]. Typically, the critical points are computed by finding all zeros of the gradient or vector field. The critical points of a scalar field are classified into minima, saddles, and maxima by the eigenvalues of its Hessian, while the critical points of a vector field are classified into sinks, saddles, and sources by the eigenvalues of its Jacobian. The respective eigenvectors can be used to compute the separatrices as the solution of an autonomous ODE. For the numerical treatment of these problems and the extraction of the periodic orbits, we refer to [18,20,5].One of the biggest challenges that such numerical algorithms face is the discrete nature of the extremal structure which necessitates a lot of binary decisions. For example, the type of a critical point depends on the sign of the eigenvalues.

Automatic, tensor-guided illustrative vector field visualization

“…Performance -our code basis has not been optimized towards performance yet. The most costly step is the persistence computation that is done for each scalar field (for details please refer to Reininghaus et al [28]). For the data sets used in the results we denote the resolution and the processing time required for the topology computation for a single scalar field on a QuadCore i7 processor with 2,6 GHz.…”

“…The same is true for the basins of all maxima. To extract the scalar field topology we employ the combinatorial framework by Reininghaus et al [28] which is robust and avoids the computation of further derivatives. Within this framework, we can use homological persistence as introduced by Edelsbrunner [8].…”

Aspects of topological approaches for data science

“…Being able to glean information about the internal structures of graphical data would be useful in solving the typical problems given to machine learning algorithms. For example, classification problems, prediction and, in particular, partitioning data into clusters [115,Section 1.1.3].…”

Topological Machine Learning for Mixed Numeric and Categorical Data

“…7,53 Topological data analysis is often referred to as studying the "shape" of data, in order to deduce fundamental characteristics of the data. The primary tool used in TDA is persistent homology, 17,56 though there are also other tools such as Mapper, 38,45 discrete Morse theory, 19,43,52 as well as other techniques from algebraic topology. 23,33,50,51 It is generally acknowledged that topological data analysis is effective at analyzing high-dimensional noisy data.…”