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Updated: Jul 18, 2025
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Mapping Cortical Dynamics Using Simultaneous MEG/EEG and Anatomically-constrained Minimum-norm Estimates: an Auditory Attention Example
Published on: October 24, 2012
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A Probabilistic Result on Impulsive Noise Reduction in Topological Data Analysis through Group Equivariant
Patrizio Frosini1, Ivan Gridelli1, Andrea Pascucci1
1Department of Mathematics, University of Bologna, 40126 Bologna, Italy.
Entropy (Basel, Switzerland)
|August 26, 2023
Summary
Group equivariant non-expansive operators (GENEOs) can now remove impulsive noise and stabilize Topological Data Analysis (TDA) with noisy data. GENEOs control persistence diagram perturbations from uniform impulsive noise in L-Lipschitz functions.
Area of Science:
- Mathematics
- Data Science
- Machine Learning
Background:
- Group equivariant non-expansive operators (GENEOs) are increasingly used in Topological Data Analysis (TDA) and Machine Learning.
- Impulsive noise and data perturbations pose challenges for TDA stability and accuracy.
Purpose of the Study:
- To investigate the application of GENEOs for impulsive noise removal in TDA.
- To demonstrate how GENEOs can enhance the stability of TDA when dealing with noisy data.
- To analyze the effect of GENEOs on persistence diagram perturbations caused by uniform impulsive noise.
Main Methods:
- Theoretical analysis of GENEOs applied to L-Lipschitz functions.
- Mathematical formulation of noise models and their impact on persistence diagrams.
- Proof of GENEOs' ability to control expected perturbation values.
Main Results:
- GENEOs are effective in removing impulsive noise from data.
- The application of GENEOs significantly increases the stability of TDA in the presence of noisy data.
- GENEOs provide control over the expected value of persistence diagram perturbations induced by uniformly distributed impulsive noise.
Conclusions:
- GENEOs offer a novel approach for robust TDA in noisy environments.
- The findings extend the applicability of GENEOs beyond their initial domains.
- This research bridges the gap between operator theory and practical data analysis challenges.
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