Abstract

In this paper, we explore the properties and optimization techniques related to polyhedral cones and energy numbers with a focus on the cone of positive semidefinite matrices and efficient computation strategies for kernels. In Part (a), we examine the polyhedral nature of the cone of positive semidefinite matrices, , establishing that it does not form a polyhedral cone for due to its infinite dimensional characteristics. In Part (b), we present an algorithm for efficiently computing the kernel function on-the-fly, leveraging a polyhedral description of the convex hull generated by the feature mappings and . By restructuring the problem and using gradient-based optimization techniques, our approach minimizes memory usage and computational overhead, thus enabling scalable computation. Through examples and visualizations, we demonstrate the practical applications and efficiency of the proposed algorithm in optimizing these kernel computations.