A Mathematical Optimization Approach to Multisphere Support Vector Data Description

2025/07/15 by Blanco, Víctor, Espejo, Inmaculada, Páez, Raúl +1
#FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2507.11106

Abstract

We present a novel mathematical optimization framework for outlier detection in multimodal datasets, extending Support Vector Data Description approaches. We provide a primal formulation, in the shape of a Mixed Integer Second Order Cone model, that constructs Euclidean hyperspheres to identify anomalous observations. Building on this, we develop a dual model that enables the application of the kernel trick, thus allowing for the detection of outliers within complex, non-linear data structures. An extensive computational study demonstrates the effectiveness of our exact method, showing clear advantages over existing heuristic techniques in terms of accuracy and robustness.

Citations

• A Unified Optimization Framework for Multiclass Classification with Structured Hyperplane Arrangements
• Anomaly Detection in Smart Power Grids with Graph-Regularized MS-SVDD: a Multimodal Subspace Learning Approach
• Newton Method-based Subspace Support Vector Data Description
• Support Vector Machines with the Hard-Margin Loss: Optimal Training via Combinatorial Benders' Cuts
• Robust Optimal Classification Trees under Noisy Labels
• Extended Isolation Forest
• Fast Incremental SVDD Learning Algorithm with the Gaussian Kernel
• Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems

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