A projected gradient solution to the minimum connector problem with extensions to support vector machines

Abstract

In this paper, we present a comprehensive study on the problem of finding the minimum connector between two convex sets, particularly focusing on polytopes, and extended to large margin classification problems. The problem holds significant relevance in diverse fields such as pattern recognition, machine learning, convex analysis, and applied linear algebra. Notably, it plays a crucial role in binary classification tasks by determining the maximum margin hyperplane that separates two sets of data. Our main contribution is the introduction of an innovative iterative approach that employs a projected gradient method to compute the minimum connector solution using only first-order information. Furthermore, we demonstrate the applicability of our method to solve the one-class problem with a single projection step, and the multi-class problem with a novel multi-objective quadratic function and a multiple projection step, which have important significance in pattern recognition and machine learning fields. Our formulation incorporates a dual representation, enabling utilization of kernel functions to address non-linearly separable problems. Moreover, we establish a connection between the solutions of the Minimum Connector and the Maximum Margin Hyperplane problems through a reparameterization technique based on collinear projection. To validate the effectiveness of our method, we conduct extensive experiments on various benchmark datasets commonly used in the field. The experimental results demonstrate the effectiveness of our approach and its ability to handle diverse applications.