# A Mathematical Optimization Approach to Multisphere Support Vector Data Description

arXiv: https://arxiv.org/abs/2507.11106 · HTML: https://mlsanity.com/paper/2507.11106/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 55 references that are in the map.
- First posted on arXiv 15 Jul 2025.

## Builds on (most cited first, 12 of 12)

- [Extended Isolation Forest](https://mlsanity.com/paper/1811.02141/) (2018)
- [Multimodal Subspace Support Vector Data Description](https://mlsanity.com/paper/1904.07698/) (2019)
- [Ellipsoidal Subspace Support Vector Data Description](https://mlsanity.com/paper/2003.09504/) (2020)
- [Graph-Embedded Subspace Support Vector Data Description](https://mlsanity.com/paper/2104.14370/) (2021)
- [Robust Optimal Classification Trees under Noisy Labels](https://mlsanity.com/paper/2012.08560/) (2020)
- [Newton Method-based Subspace Support Vector Data Description](https://mlsanity.com/paper/2309.13960/) (2023)
- [A robust SVM-based approach with feature selection and outliers detection for classification problems](https://mlsanity.com/paper/2403.07753/) (2024)
- [On the multisource hyperplanes location problem to fitting set of points](https://mlsanity.com/paper/2002.01692/) (2020)
- [Support Vector Machines with the Hard-Margin Loss: Optimal Training via Combinatorial Benders' Cuts](https://mlsanity.com/paper/2207.07690/) (2022)
- [Tightening big Ms in integer programming formulations for support vector machines with ramp loss](https://mlsanity.com/paper/2403.07736/) (2024)
- [Fast Incremental SVDD Learning Algorithm with the Gaussian Kernel](https://mlsanity.com/paper/1709.00139/) (2017)
- [Anomaly Detection in Smart Power Grids with Graph-Regularized MS-SVDD: a Multimodal Subspace Learning Approach](https://mlsanity.com/paper/2502.15793/) (2025)

## Built on by (most cited first, 1 of 1)

- [Multi-Modal Anomaly Detection: A Survey](https://mlsanity.com/paper/2608.24937/) (2026)

## Abstract (arXiv)

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.