Unlocking the Potential of Non-PSD Kernel Matrices: A Polar Decomposition-based Transformation for Improved Prediction Models

Kernel functions are a key element in many machine learning methods to capture the similarity between data points. However, a considerable number of these functions do not meet all mathematical requirements to be a valid positive semi-definite kernel, a crucial precondition for kernel-based classifiers such as Support Vector Machines or Kernel Fisher Discriminant classifiers. In this paper, we propose a novel strategy employing a polar decomposition to effectively transform invalid kernel matrices to positive semi-definite matrices, while preserving the topological structure inherent to the data points. Utilizing polar decomposition allows the effective transformation of indefinite kernel matrices from Krein space to positive semi-definite matrices in Hilbert space, thereby providing an efficient out-of-sample extension for new unseen data and enhancing kernel method applicability across diverse classification tasks. We evaluate our approach on a variety of benchmark datasets and demonstrate its superiority over competitive methods.