4,123 research outputs found
Distributed computation of persistent homology
Persistent homology is a popular and powerful tool for capturing topological
features of data. Advances in algorithms for computing persistent homology have
reduced the computation time drastically -- as long as the algorithm does not
exhaust the available memory. Following up on a recently presented parallel
method for persistence computation on shared memory systems, we demonstrate
that a simple adaption of the standard reduction algorithm leads to a variant
for distributed systems. Our algorithmic design ensures that the data is
distributed over the nodes without redundancy; this permits the computation of
much larger instances than on a single machine. Moreover, we observe that the
parallelism at least compensates for the overhead caused by communication
between nodes, and often even speeds up the computation compared to sequential
and even parallel shared memory algorithms. In our experiments, we were able to
compute the persistent homology of filtrations with more than a billion (10^9)
elements within seconds on a cluster with 32 nodes using less than 10GB of
memory per node
Discrete Morse theory for computing cellular sheaf cohomology
Sheaves and sheaf cohomology are powerful tools in computational topology,
greatly generalizing persistent homology. We develop an algorithm for
simplifying the computation of cellular sheaf cohomology via (discrete)
Morse-theoretic techniques. As a consequence, we derive efficient techniques
for distributed computation of (ordinary) cohomology of a cell complex.Comment: 19 pages, 1 Figure. Added Section 5.
Computing multiparameter persistent homology through a discrete Morse-based approach
Persistent homology allows for tracking topological features, like loops, holes and their higher-dimensional analogues, along a single-parameter family of nested shapes. Computing descriptors for complex data characterized by multiple parameters is becoming a major challenging task in several applications, including physics, chemistry, medicine, and geography. Multiparameter persistent homology generalizes persistent homology to allow for the exploration and analysis of shapes endowed with multiple filtering functions. Still, computational constraints prevent multiparameter persistent homology to be a feasible tool for analyzing large size data sets. We consider discrete Morse theory as a strategy to reduce the computation of multiparameter persistent homology by working on a reduced dataset. We propose a new preprocessing algorithm, well suited for parallel and distributed implementations, and we provide the first evaluation of the impact of multiparameter persistent homology on computations
Cellular Sheaves And Cosheaves For Distributed Topological Data Analysis
This dissertation proposes cellular sheaf theory as a method for decomposing data analysis problems. We present novel approaches to problems in pursuit and evasion games and topological data analysis, where cellular sheaves and cosheaves are used to extract global information from data distributed with respect to time, boolean constraints, spatial location, and density. The main contribution of this dissertation lies in the enrichment of a fundamental tool in topological data analysis, called persistent homology, through cellular sheaf theory. We present a distributed computation mechanism of persistent homology using cellular cosheaves. Our construction is an extension of the generalized Mayer-Vietoris principle to filtered spaces obtained via a sequence of spectral sequences. We discuss a general framework in which the distribution scheme can be adapted according to a user-specific property of interest. The resulting persistent homology reflects properties of the topological features, allowing the user to perform refined data analysis. Finally, we apply our construction to perform a multi-scale analysis to detect features of varying sizes that are overlooked by standard persistent homology
Parallel decomposition of persistence modules through interval bases
We introduce an algorithm to decompose any finite-type persistence module
with coefficients in a field into what we call an {em interval basis}. This
construction yields both the standard persistence pairs of Topological Data
Analysis (TDA), as well as a special set of generators inducing the interval
decomposition of the Structure theorem. The computation of this basis can be
distributed over the steps in the persistence module. This construction works
for general persistence modules on a field , not necessarily
deriving from persistent homology. We subsequently provide a parallel algorithm
to build a persistent homology module over by leveraging the Hodge
decomposition, thus providing new motivation to explore the interplay between
TDA and the Hodge Laplacian
Parallel decomposition of persistence modules through interval bases
We introduce an algorithm to decompose any finite-type persistence module
with coefficients in a field into what we call an {\em interval basis}. This
construction yields both the standard persistence pairs of Topological Data
Analysis (TDA), as well as a special set of generators inducing the interval
decomposition of the Structure theorem. The computation of this basis can be
distributed over the steps in the persistence module. This construction works
for general persistence modules on a field , not necessarily
deriving from persistent homology. We subsequently provide a parallel algorithm
to build a persistent homology module over by leveraging the Hodge
decomposition, thus providing new motivation to explore the interplay between
TDA and the Hodge Laplacian.Comment: 37 pages, 6 figure
Distributing Persistent Homology via Spectral Sequences
We set up the theory for a distributed algorithm for computing persistent
homology. For this purpose we develop linear algebra of persistence modules. We
present bases of persistence modules, and give motivation as for the advantages
of using them. Our focus is on developing efficient methods for the computation
of homology of chains of persistence modules. Later we give a brief, self
contained presentation of the Mayer-Vietoris spectral sequence. Then we study
the Persistent Mayer-Vietoris spectral sequence and present a solution to the
extension problem. Finally, we review PerMaViss, a method implementing these
ideas. This procedure distributes simplicial data, while focusing on merging
homological information.Comment: Comments: 31 pages, 14 figures, 1 algorithm. Changes to previous
version: longer introduction added, some material has been removed due to
length constraints, section 5.2 added describing the procedure of computing
the persistence Mayer-Vietoris spectral sequence, followed by complexity
estimates in section 5.
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