A Mathematical Optimization Approach to Multisphere Support Vector Data Description

2025-07-15

AI Summary

This paper addresses anomaly detection in multimodal data by proposing a mathematically optimized Multi-Sphere Support Vector Data Description (MSVDD) framework. The method models normal samples using multiple Euclidean hyperspheres and incorporates kernel tricks to capture nonlinear structures. It introduces, for the first time, both primal and dual Mixed-Integer Second-Order Cone Programming (MISOCP) formulations, enabling globally optimal and exact solutions—overcoming the local optima limitations of conventional heuristic algorithms. Theoretical analysis integrates duality theory with kernel mapping to ensure model interpretability and generalization guarantees. Extensive experiments on benchmark multimodal datasets demonstrate that MSVDD significantly outperforms state-of-the-art methods in detection accuracy, robustness, and stability.

Technology Category

Machine Learning: Multimodal LearningComputer Vision: Multi-modal VisionData Mining & Knowledge Management: Anomaly/Outlier Detection

Application Category

Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web dataGraph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphsSearch and Retrieval-Augmented AI: Web query analysis, representation and understanding

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.

Problem

• Extends SVDD for outlier detection in multimodal datasets
• Develops Mixed Integer Second Order Cone model for anomalies
• Enables kernel trick for non-linear outlier detection

Innovation

• Mixed Integer Second Order Cone model
• Dual model enabling kernel trick
• Exact method for outlier detection