A topological classifier to characterize brain states: When shape matters more than variance

Abstract: Despite the remarkable accuracies attained by machine learning classifiers to separate complex datasets in a supervised fashion, most of their operation falls short to provide an informed intuition about the structure of data, and, what is more important, about the phenomena being characterized by the given datasets. By contrast, topological data analysis (TDA) is devoted to study the shape of data clouds by means of persistence descriptors and provides a quantitative characterization of specific topological features of the dataset under scrutiny. In this article we introduce a novel TDA-based classifier that works on the principle of assessing quantifiable changes on topological metrics caused by the addition of new input to a subset of data. We used this classifier with a high-dimensional electro-encephalographic (EEG) dataset recorded from eleven participants during a decision-making experiment in which three motivational states were induced through a manipulation of social pressure. After processing a band-pass filtered version of EEG signals, we calculated silhouettes from persistence diagrams associated with each motivated state, and classified unlabeled signals according to their impact on each reference silhouette. Our results show that in addition to providing accuracies within the range of those of a nearest neighbour classifier, the TDA classifier provides formal intuition of the structure of the dataset as well as an estimate of its intrinsic dimension. Towards this end, we incorporated dimensionality reduction methods to our procedure and found that the accuracy of our TDA classifier is generally not sensitive to explained variance but rather to shape, contrary to what happens with most machine learning classifiers.

• The paper presents a TDA-based classifier that leverages persistence diagrams to capture data shape over variance for EEG analysis.
• The methodology transforms persistence diagrams into silhouettes, comparing topological distances to achieve competitive accuracy with traditional classifiers.
• Experimental results on EEG data under social pressure reveal enhanced performance in reduced-dimensional γ frequency bands.

A Topological Classifier to Characterize Brain States: When Shape Matters More than Variance

Introduction

The paper "A topological classifier to characterize brain states: When shape matters more than variance" (2303.04231) introduces a novel approach to data classification using Topological Data Analysis (TDA), diverging from traditional machine learning paradigms. Instead of focusing on the variance within datasets, the proposed TDA-based classifier leverages the shape and structure of point clouds in high-dimensional spaces. This technique is applied to electroencephalographic (EEG) data recorded during a decision-making experiment involving social pressure-induced motivational states.

Persistence Summaries and Classifier Design

TDA, particularly persistent homology, provides a framework for capturing the topological features of datasets, such as connectivity and cycles, at various resolution scales. The classifier operates by evaluating changes in topological metrics, using persistence diagrams and landscapes as primary tools. These diagrams are transformed into persistence silhouettes, which serve as summaries for assessing the shape of class-specific data clouds.

The classifier's unique methodology involves comparing the persistence silhouettes of different classes and calculating distances between them. When a new data point is added to a class, minimal changes in topological descriptors indicate correct classification, whereas notable changes suggest misclassification.

Figure 1: Persistence landscape obtained from a persistence diagram by means of 45-degree rotation and rescaling.

Figure 2: Tent function for a birth parameter $b$ and a death parameter $d$.

Experimental Setup with EEG Data

The classification task utilized high-dimensional EEG recordings from eleven participants engaged in a decision-making task affected by social pressure. The signals were processed to explore the motivational impact of three conditions: solo, easy, and hard, each corresponding to varying levels of social pressure. Data preprocessing involved filtering the EEG signals into typical frequency bands ($\alpha$, $\beta$, $\gamma$) and performing a dimensionality reduction using methods like PCA and RFE.

Figure 3: Silhouettes from persistence diagrams in dimension zero for each motivational state for each frequency band and unfiltered dataset.

Figure 4: Pre-processing schematic showing data filtering and classification setup.

Results: Influence of Dimensionality Reduction and Frequency Bands

The study assessed classification accuracies across source and electrode spaces, employing different dimensionality reduction techniques. It was found that TDA classifier performance peaked within reduced dimensionality spaces, particularly in the $\gamma$ frequency band, revealing the classifier's sensitivity to shape rather than variance.

Figure 5: Accuracies of the topological classifier on source space by frequency band without dimensionality reduction for selected participants.

Figure 6: Comparison of baseline accuracies with those obtained after dimensionality reduction via PCA and RFE for participants.

Discussion

The novel TDA-based classifier provides comparable accuracies to conventional methods like nearest neighbor classification (1-NN). However, its reliance on shape characteristics rather than explained variance highlights a critical difference in approach. While 1-NN classifiers exhibit improved performance with increased variance explanation, TDA classifiers prioritize intrinsic data shape, which stabilizes at a specific dimension.

This behavior suggests potential applications where interpretability and data structure insights are crucial, such as understanding neural states during complex cognitive tasks. The findings reinforce the effectiveness of TDA in capturing topological features relevant for classification, particularly in high-dimensional biological data.

Conclusion

The study presents a promising avenue for data classification using topological analyses, advancing both theoretical understanding and practical application in neuroscience. The TDA classifier's focus on shape over variance offers insights into data structure that traditional methods may overlook, opening possibilities for more interpretable AI systems in sensitive applications. Further research could explore expanding the classifier's applicability across varied datasets and refining its robustness in different domains.