Persistent sheaf cohomology

Lead Research Organisation: CARDIFF UNIVERSITY
Department Name: Sch of Mathematics


Algebraic topology studies the features of topological spaces which are invariant under continuous deformations, like the number of holes in a surface or the Euler characteristic of a polyhedron. In data analysis and computer science on the other hand geometric objects are often approximated by ""point clouds"", which rises the following question: How can we reconstruct the topological invariants of a closed subset of Euclidean space from a set of possibly inaccurate point samples? There is a way to associate a topological space to such a point cloud, which is based on a threshold radius and is called the Vietoris-Rips complex. Persistent Homology captures the development of the homology groups of these complexes for all values of the threshold radius in so-called persistence diagrams, which allow conclusions about the invariants of the original space.

The project is about a fairly recent development: the application of sheaf theory to topological data analysis. Sheaves are used to described data distributed over a topological space and to solve ""local to global"" problems. The project will contribute to the development of the theoretical groundwork for these applications by generalising results in persistent cohomology, like stability theorems, to the sheaf theoretic context.


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Torras-Casas Á (2023) Distributing Persistent Homology via Spectral Sequences in Discrete & Computational Geometry

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509449/1 30/09/2016 29/09/2021
1941653 Studentship EP/N509449/1 30/09/2017 30/03/2021 Alvaro Torras Casas
Description 'Topology' is an area of mathematics that studies the shape of objects, focusing on their most essential properties. 'Topological Data Analysis' (TDA) is an emerging area of research that applies methods and techniques from Topology to problems that appear in applied disciplines. A central tool in TDA is Persistent Homology (PH), which is used to describe complex information about n-dimensional holes in terms of a barcode. Among others, PH has been applied to detect pulmonary disease, study coverage in sensor networks, material classification, compressibility of molecules and shape recognition using machine learning techniques.

However, PH is very expensive, both in computational time and memory required. This is why our research has been focused on bringing in more recent ideas from Topology, in order to improve on the current methods. In particular, we have adapted the Mayer-Vietoris spectral sequence for the distributed computation of PH. In a nutshell, the idea is to break down the data into parts, compute PH for each separate component and recompose the result using the spectral sequence. However, this approach has some theoretical obstructions. The main difficulty relies on the fact that the resulting barcode might be 'broken down' into pieces, and it is not straightforward to recompose them. These results can be found on the preprint: arXiv:1907.05228 [math.AT]

Another relevant contribution is our first release of PerMaViss, a python software that computes PH by using the aforementioned method. Right now, this package is a 'proof of concept' for the correctness of our approach. Further, one can read off more information than the standard PH barcode from the PerMaViss computations. This solves two problems at the same time. On one hand, we are developing a distributive algorithm for PH. On the other, we get more understanding for the meaning of the barcodes. Altogether, this will contribute to the impact of TDA in areas such as neuroscience, machine learning, and more.
Exploitation Route Since we have developed a new tool for analysing the shape of data, there are a number of both theoretical and practical questions. These include how to optimize PerMaViss, and how to use the extra information given by our method. This could be of interest for research groups that have already used topology, such as the 'Blue Brain Project' in Switzerland, or the 'centre for Topological Data Analysis' in Liverpool, Swansea and Oxford. Apart from academic questions, there is a potential for industrial impact as well. Currently, Giotto and Ayasdi are two start-ups applying topological methods for problems arising in the financial, public and health sectors.
Sectors Chemicals,Digital/Communication/Information Technologies (including Software),Financial Services, and Management Consultancy,Healthcare,Manufacturing, including Industrial Biotechology,Pharmaceuticals and Medical Biotechnology,Other

Title PerMaViss: Persistence Mayer Vietoris spectral sequence 
Description This library is intended to be a proof of concept for Persistence Homology (PH) parallelization. That is, one can divide a point cloud into covering regions, compute PH on each part, and combine all results to obtain the global PH again. This is done by means of the Persistence Mayer Vietoris spectral sequence. The method comes from the mathematical ideas contained in "Distributing Persistent Homology via Spectral Sequences" Álvaro Torras Casas, arXiv:1907.05228 [math.AT] The library contains two key examples, the torus and random point clouds in three dimensions. Both of these are divided into 8 mutually overlapping regions, and the spectral sequence is computed with respect to this cover. The resulting barcodes coincide with that which would be obtained by computing PH directly. 
Type Of Technology Software 
Year Produced 2020 
Open Source License? Yes  
Impact This library proves that spectral sequences can be used for computing Persistent Homology. Furthermore, one can see that this method brings extra information to Persistent Homology, leading to new ways to understand the shape of datasets. This has been a long-standing problem in the area of Topological Data Analysis, where some researchers suggested that this approach could be used to parallelize computations. This will contribute to the impact that Persistent Homology already had on areas such as medicine, chemistry and neuroscience. Right now the current implementation PerMaViss(version 0.0.2) still needs to be optimized. Also, it would be interesting to implement and test it on a cluster.