Application on Geometric Machine Learning

Abstract

The geometrization of machine learning problems is driven by the need to process big data. For the application of ML methods, the data needs to be placed in a space that allows processing of these data. Geometric machine learning (GML) is used to develop an ML model that unifies ML with a geometric data model. This paper explores geometric ML models for different types of data. This work focuses on the generalization of transformers in graph neural networks (GNNs) and the time-series data regression model. We first address the issues of explainability and expressiveness in graph neural networks and study the geometry of the multihead attention mechanism.

In our study, we suggest a fusion of deep neural networks and fuzzy modeling principles for feature representation learning. Two critical challenges demand our focus in this framework. Firstly, the exponential increase in fuzzy rules tied to the number of features poses computational inefficiencies. Secondly, the complexity of the division subspace, stemming from the integration of fuzzy rules, gives rise to nonlinear relationships between dependent and independent variables. To overcome these hurdles, we present a feature sampling method and employ simulations utilizing diverse fitting curves based on polynomial functions.

In addition, the perspective of geometric understanding applies to standard regression problems, and the prediction of time series data provides a clear insight into the geometry of classifiers. Time-series analysis involves measuring the similarity between two sequences that may differ in speed. During the prediction process, geometric distortion of two-time series data may occur. The predicted value may not match the observed sequences according to the exact time index. Therefore, weighting the predicted difference to observations from one-to-one index points or different one-to-one time indexes is necessary and practical. Integrating geometric synchronization into time series analysis is another major goal of this work. Finally, all methodologies are ready for several real-world applications. We address the problem of fetal monitoring for early abnormal detection problems.

The goals of the thesis are the following.

\begin{enumerate} \item Investigate and study transformer and graph neural networks and conceive of a graph-specific attention mechanism from a geometric perspective. \item Research the geometric understanding of feature representation and incorporate the fuzzy concept into feature learning. \item Investigate a time-warping tool for geometric prototype synchronization and employ the principle of machine learning for objective optimization. \item Organize and summarize current research gaps and issues in fetal monitoring. Then, propose a practical framework for the general application. \end{enumerate} }