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Optimising the topological information of the A_∞-persistence groups
Persistent homology typically studies the evolution of homology groups H...
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A Fast and Robust Method for Global Topological Functional Optimization
Topological statistics, in the form of persistence diagrams, are a class...
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Approximate Nearest Neighbors in the Space of Persistence Diagrams
Persistence diagrams are important tools in the field of topological dat...
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Optimizing persistent homology based functions
Solving optimization tasks based on functions and losses with a topologi...
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A Progressive Approach to Scalar Field Topology
This paper introduces progressive algorithms for the topological analysi...
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Persistent Homology of Complex Networks for Dynamic State Detection
In this paper we develop a novel Topological Data Analysis (TDA) approac...
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Localized Topological Simplification of Scalar Data
This paper describes a localized algorithm for the topological simplific...
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Topological Regularization via Persistence-Sensitive Optimization
Optimization, a key tool in machine learning and statistics, relies on regularization to reduce overfitting. Traditional regularization methods control a norm of the solution to ensure its smoothness. Recently, topological methods have emerged as a way to provide a more precise and expressive control over the solution, relying on persistent homology to quantify and reduce its roughness. All such existing techniques back-propagate gradients through the persistence diagram, which is a summary of the topological features of a function. Their downside is that they provide information only at the critical points of the function. We propose a method that instead builds on persistence-sensitive simplification and translates the required changes to the persistence diagram into changes on large subsets of the domain, including both critical and regular points. This approach enables a faster and more precise topological regularization, the benefits of which we illustrate with experimental evidence.
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