A Mathematical Optimization Approach to Multisphere Support Vector Data Description

TL;DR

This paper introduces a new mathematical optimization framework for outlier detection in complex datasets using support vector data description, leveraging mixed integer second order cone programming and kernel methods for improved accuracy.

Contribution

It develops a primal and dual model for outlier detection that extends existing SVM-based methods with a rigorous optimization approach and kernelization capabilities.

Findings

Exact method outperforms heuristics in accuracy.

Framework effectively detects outliers in non-linear data.

Demonstrates robustness across diverse datasets.

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.