| dc.description.abstract | The unique functioning of Support Vector Machines (SVMs) can create undesirable behaviour, however this unique functioning in turn also allow for exploration of novel topological regularization methods. SVMs are a common machine learning model used for classification tasks. Like most machine learning models, SVMs can be prone to overfitting to training data, which hurts its generalization. When optimizing the generalization of classifiers we can consider topological properties of the decision boundaries. Intuitively, a decision boundary slicing the input space into numerous disjoint components is less likely to generalize effectively. Building on this assumption, I propose two novel topological generalization techniques using properties specific to SVMs. SVMs work by mapping data into a higher-dimensional feature space, with the goal of making data linear separable among classes. Analysing this feature space with methods from topology and Morse theory leads to new methods to change an SVM's decision boundary in a manner that reduces its topological complexity. In particular, I define a measure of topological complexity for SVMs and propose 2 methods intended to decrease this complexity. I show practical utility of these methods through illustrative examples, which indicate effectiveness in real-world scenarios. Moreover, the results of experiments challenge the idea that a maximal margin results in an optimal decision boundary. This thesis's goal is to serve as a basic exploration into topological regularization for SVMs, providing ideas for further research. In the discussion I identify weaknesses and challenges such as reliance on approximations and computationally costly algorithms, which can be tackled through further future research. | |
