Reformulation of the SVM+ optimization problem into quadratic programming form

Abstract

Support Vector Machines with Privileged Information (SVM+), developed within the Learning with Privileged Information (LUPI) paradigm, improve upon classical SVMs by integrating additional information available only during training, thereby improving generalization. However, this extra information increases the complexity of the associated optimization problem. Unlike classical SVMs, where highly optimized algorithms like SMO exist, SVM+ lacks a standardized formulation that allows leveraging modern optimization solvers.

In this work, the SVM+ dual problem is explicitly reformulated into a standard matrix Quadratic Programming (QP) form. This transformation allows delegating the problem resolution to robust and highly efficient numerical solvers (such as MATLAB's quadprog or industrial optimizers), avoiding the need to implement complex ad-hoc iterative algorithms. The mathematical equivalence between the original formulation and the proposed QP is rigorously validated using a symbolic resolution engine (symbolic solve) as Ground Truth, demonstrating that the reformulation preserves numerical accuracy (error <10⁻⁸) while drastically reducing computation times compared to generic optimizers. This methodology renders SVM+ computationally accessible for larger-scale supervised learning tasks in scientific and industrial environments.

This formulation is particularly valuable in industrial and scientific settings where high-dimensional privileged information, such as expert annotations or offline measurements, is utilized. A notable application is the prediction of plasma disruptions in nuclear fusion devices, where classification performance is enhanced by leveraging offline diagnostic data. Consequently, by providing a standardized and efficient optimization structure, SVM+ becomes practically viable for challenging, real-world supervised machine learning tasks.