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Learning Simplicial Complexes from Persistence Diagrams
Topological Data Analysis (TDA) studies the shape of data. A common topo...
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A Progressive Approach to Scalar Field Topology
This paper introduces progressive algorithms for the topological analysi...
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GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodes
The computation of Vietoris-Rips persistence barcodes is both execution-...
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Topological Regularization via Persistence-Sensitive Optimization
Optimization, a key tool in machine learning and statistics, relies on r...
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Topologically Controlled Lossy Compression
This paper presents a new algorithm for the lossy compression of scalar ...
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A note on stochastic subgradient descent for persistence-based functionals: convergence and practical aspects
Solving optimization tasks based on functions and losses with a topologi...
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GPU Parallel Computation of Morse-Smale Complexes
The Morse-Smale complex is a well studied topological structure that rep...
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Localized Topological Simplification of Scalar Data
This paper describes a localized algorithm for the topological simplification of scalar data, an essential pre-processing step of topological data analysis (TDA). Given a scalar field f and a selection of extrema to preserve, the proposed localized topological simplification (LTS) derives a function g that is close to f and only exhibits the selected set of extrema. Specifically, sub- and superlevel set components associated with undesired extrema are first locally flattened and then correctly embedded into the global scalar field, such that these regions are guaranteed – from a combinatorial perspective – to no longer contain any undesired extrema. In contrast to previous global approaches, LTS only and independently processes regions of the domain that actually need to be simplified, which already results in a noticeable speedup. Moreover, due to the localized nature of the algorithm, LTS can utilize shared-memory parallelism to simplify regions simultaneously with a high parallel efficiency (70 the exploration of simplification parameters and their effect on subsequent topological analysis. For such exploration tasks, LTS brings the overall execution time of a plethora of TDA pipelines from minutes down to seconds, with an average observed speedup over state-of-the-art techniques of up to x36. Furthermore, in the special case where preserved extrema are selected based on topological persistence, an adapted version of LTS partially computes the persistence diagram and simultaneously simplifies features below a predefined persistence threshold. The effectiveness of LTS, its parallel efficiency, and its resulting benefits for TDA are demonstrated on several simulated and acquired datasets from different application domains, including physics, chemistry, and biomedical imaging.
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