Towards Convexity in Anomaly Detection: A New Formulation of SSLM with Unique Optimal Solutions

TL;DR

This paper introduces a convex formulation of SSLM for anomaly detection, enabling better analysis of solutions, hyperparameter effects, and uniqueness, which was difficult with traditional nonconvex methods.

Contribution

A novel convex SSLM formulation that simplifies analysis, guarantees unique solutions, and clarifies hyperparameter impacts compared to traditional nonconvex approaches.

Findings

Convex SSLM reduces to quadratic programming for key hyperparameters.

Analysis of hyperparameter influence on optimal solutions.

Identification of conditions for solution uniqueness.

Abstract

An unsolved issue in widely used methods such as Support Vector Data Description (SVDD) and Small Sphere and Large Margin SVM (SSLM) for anomaly detection is their nonconvexity, which hampers the analysis of optimal solutions in a manner similar to SVMs and limits their applicability in large-scale scenarios. In this paper, we introduce a novel convex SSLM formulation which has been demonstrated to revert to a convex quadratic programming problem for hyperparameter values of interest. Leveraging the convexity of our method, we derive numerous results that are unattainable with traditional nonconvex approaches. We conduct a thorough analysis of how hyperparameters influence the optimal solution, pointing out scenarios where optimal solutions can be trivially found and identifying instances of ill-posedness. Most notably, we establish connections between our method and traditional approaches, providing a clear determination of when the optimal solution is unique--a task unachievable with traditional nonconvex methods. We also derive the nu-property to elucidate the interactions between hyperparameters and the fractions of support vectors and margin errors in both positive and negative classes.