Computational topology | EPFL Graph Search

Algorithmic topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular, computational geometry and computational complexity theory. A primary concern of algorithmic topology, as its name suggests, is to develop efficient algorithms for solving problems that arise naturally in fields such as computational geometry, graphics, robotics, structural biology and chemistry, using methods from computable topology. A large family of algorithms concerning 3-manifolds revolve around normal surface theory, which is a phrase that encompasses several techniques to turn problems in 3-manifold theory into integer linear programming problems. Rubinstein and Thompson's 3-sphere recognition algorithm. This is an algorithm that takes as input a triangulated 3-manifold and determines whether or not the manifold is homeomorphic to the 3-sphere. It has exponential run-time in the number of tetrahedral simplexes in the initial 3-manifold, and also an exponential memory profile. Moreover, it is implemented in the software package Regina. Saul Schleimer went on to show the problem lies in the complexity class NP. Furthermore, Raphael Zentner showed that the problem lies in the complexity class coNP, provided that the generalized Riemann hypothesis holds. He uses instanton gauge theory, the geometrization theorem of 3-manifolds, and subsequent work of Greg Kuperberg on the complexity of knottedness detection. The connect-sum decomposition of 3-manifolds is also implemented in Regina, has exponential run-time and is based on a similar algorithm to the 3-sphere recognition algorithm. Determining that the Seifert-Weber 3-manifold contains no incompressible surface has been algorithmically implemented by Burton, Rubinstein and Tillmann and based on normal surface theory. The Manning algorithm is an algorithm to find hyperbolic structures on 3-manifolds whose fundamental group have a solution to the word problem. At present the JSJ decomposition has not been implemented algorithmically in computer software.

Reduced representations of complexes, signals, and multifiltrations

Celia Camille Hacker

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

Recompression Of Hadamard Products Of Tensors In Tucker Format

Daniel Kressner, Lana Perisa

The Hadamard product features prominently in tensor-based algorithms in scientific computing and data analysis. Due to its tendency to significantly increase ranks, the Hadamard product can represent a major computational obstacle in algorithms based on low-rank tensor representations. It is therefore of interest to develop recompression techniques that mitigate the effects of this rank increase. In this work, we investigate such techniques for the case of the Tucker format, which is well suited for tensors of low order and small to moderate multilinear ranks. Fast algorithms are attained by combining iterative methods, such as the Lanczos method and randomized algorithms, with fast matrix-vector products that exploit the structure of Hadamard products. The resulting complexity reduction is particularly relevant for tensors featuring large mode sizes I and small to moderate multilinear ranks R. To implement our algorithms, we have created a new Julia library for tensors in Tucker format.

Siam Publications2017

Reduced representations of complexes, signals, and multifiltrations

Celia Camille Hacker

EPFL2022