Discrete Morse theory

Summary

Discrete Morse theory is a combinatorial adaptation of Morse theory developed by Robin Forman. The theory has various practical applications in diverse fields of applied mathematics and computer science, such as configuration spaces, homology computation, denoising, mesh compression, and topological data analysis. Notation regarding CW complexes Let X be a CW complex and denote by \mathcal{X} its set of cells. Define the incidence function \kappa\colon\mathcal{X} \times \mathcal{X} \to \mathbb{Z} in the following way: given two cells \sigma and \tau in \mathcal{X}, let \kappa(\sigma,~\tau) be the degree of the attaching map from the boundary of \sigma to \tau. The boundary operator is the endomorphism \partial of the free abelian group generated by \mathcal{X} defined by :\partial(\sigma) = \sum_{\tau \in \mathcal{X}}\kappa(\s

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https://en.wikipedia.org/wiki/Discrete_Morse_theory

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The field of computational topology has developed many powerful tools to describe the shape of data, offering an alternative point of view from classical statistics. This results in a variety of complex structures that are not always directly amenable for machine learning tasks. We develop theory and algorithms to produce computable representations of simplicial or cell complexes, potentially equipped with additional information such as signals and multifiltrations. The common goal of the topics discussed in this thesis is to find reduced representations of these often high dimensional and complex structures to better visualize, transform or formulate theoretical results about them. We extend the well known graph learning algorithm node2vec to simplicial complexes, a higher dimensional analogue of graphs. To this end we propose a way to define random walks on simplicial complexes, which we then use to design an extension of node2vec called k-simplex2vec, producing a representation of the simplices in a Euclidean space. Furthermore, the study of this method leads to interesting questions about robustness of graph and simplicial learning methods. In the case of graphs, we study node2vec embeddings arising from different parameter sets, analysing their quality and stability using various measures. In the topic of signal processing, we explore how discrete Morse theory can be used for compression and reconstruction of cell complexes equipped with signals. In particular we study the effect of the compression of a complex on the Hodge decomposition of its signals. We study how the signal changes through compression and reconstruction by introducing a topological reconstruction error, showing in particular that part of the Hodge decomposition is preserved. Moreover, we prove that any deformation retract over R can be expressed as a Morse deformation retract in a well-chosen basis, thus extending the reconstruction results to any deformation retract. In addition, we introduce an algorithm to minimize the loss induced by the reconstruction of a compressed signal. Finally, we use discrete Morse theory to compute an invariant of multi-parameter persistent homology, the rank invariant. We can restrict a multi-parameter persistence module to a one- dimensional persistence module along any line of positive slope and compute the one-dimensional analogue of the rank invariant, namely the barcode. Through a discrete Morse matching we can determine critical values in the multifiltration, which in turn allows us to identify equivalence classes of lines in the parameter space. In our main result, we explain how to compute the barcode along any given line of an equivalence class given the barcode along a representative line. This provides a way to fiber the rank invariant according to the critical values of a discrete Morse matching and to perform computations in the corresponding one-dimensional module, which is much better understood.

EPFL2022

Space discretization for efficient human navigation

Daniel Thalmann

There is a large body of research on motion control of legs in human models. However, they require specification of global paths in which to move. A method for automatically computing a global motion path for a human in a 3D environment of obstacles is presented. Object space is discretized into a 3D grid of uniform cells and an optimal path is generated between two points as a discrete cell path. The grid is treated as graph with orthogonal links of uniform cost. The A* search method is applied for path finding. By considering only the cells on the upper surface of objects on which a human walks, a large portion of the grid is discarded from the search space, thus boosting efficiency. This is expected to be a higher level mechanism for various local foot placement methods in human animation

1998

EPFL2022