🍩 Database of Original & Non-Theoretical Uses of Topology

1. Geometric Feature Performance Under Downsampling for EEG Classification Tasks (2021)

   Bryan Bischof, Eric Bunch

   Abstract

   We experimentally investigate a collection of feature engineering pipelines for use with a CNN for classifying eyes-open or eyes-closed from electroencephalogram (EEG) time-series from the Bonn dataset. Using the Takens' embedding--a geometric representation of time-series--we construct simplicial complexes from EEG data. We then compare \$\epsilon\$-series of Betti-numbers and \$\epsilon\$-series of graph spectra (a novel construction)--two topological invariants of the latent geometry from these complexes--to raw time series of the EEG to fill in a gap in the literature for benchmarking. These methods, inspired by Topological Data Analysis, are used for feature engineering to capture local geometry of the time-series. Additionally, we test these feature pipelines' robustness to downsampling and data reduction. This paper seeks to establish clearer expectations for both time-series classification via geometric features, and how CNNs for time-series respond to data of degraded resolution.

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   • Classification neural networks
   • Machine learning
   • Neuroscience
   • EEG
   • Electric signals
   • Electroencephalogram
   • Time series
   • ε-series of Betti-numbers
   • ε-series of graph spectra

2. Topological Data Analysis of Single-Trial Electroencephalographic Signals (2018)

   Yuan Wang, Hernando Ombao, Moo K. Chung

   Abstract

   Epilepsy is a neurological disorder that can negatively affect the visual, audial and motor functions of the human brain. Statistical analysis of neurophysiological recordings, such as electroencephalogram (EEG), facilitates the understanding and diagnosis of epileptic seizures. Standard statistical methods, however, do not account for topological features embedded in EEG signals. In the current study, we propose a persistent homology (PH) procedure to analyze single-trial EEG signals. The procedure denoises signals with a weighted Fourier series (WFS), and tests for topological difference between the denoised signals with a permutation test based on their PH features persistence landscapes (PL). Simulation studies show that the test effectively identifies topological difference and invariance between two signals. In an application to a single-trial multichannel seizure EEG dataset, our proposed PH procedure was able to identify the left temporal region to consistently show topological invariance, suggesting that the PH features of the Fourier decomposition during seizure is similar to the process before seizure. This finding is important because it could not be identified from a mere visual inspection of the EEG data and was in fact missed by earlier analyses of the same dataset.

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   • Epilepsy
   • Medicine
   • Electroencephalogram
   • Persistence landscape
   • Persistent homology
   • Weighted Fourier series