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

TL;DR

This paper addresses nonconvexity in hypersphere-based anomaly detection models such as SSLM and SVDD by proposing a convex kernelized SSLM (CSSLM). For fixed hyperparameters, CSSLM reduces to a convex quadratic or linear program, enabling a global optimum and enabling rigorous analysis of existence, uniqueness, and dual structure, including a novel $\nu$-property that ties hyperparameters to support-vector and margin behavior. The authors establish conditions under which the optimal sphere center is unique and characterize the radius and margin via dual variables, linking CSSLM to traditional SSLM and SVDD as special cases. Empirical results on several datasets show CSSLM consistently achieves strong performance and stable convergence, while nonconvex SSLM often yields local optima and dual SSLM may fail to align with the primal optimum. The work highlights the practical impact of convex reformulations for scalable, reliable anomaly detection and opens avenues for efficient large-scale optimization and further theoretical properties.

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.

Figures (9)

• Figure 1: The flowchart of CSSLM.
• Figure 2: the optimal $r_*$ and $t_*$ of \ref{['prob.unc']} dependent on different choices of $\mu$ and $\nu$.
• Figure 3: Connection to SSLM and SVDD
• Figure 4: Optimal values of the primal SSLM \ref{['prob.sslm']} under 100 random initializations on the Banana Dataset (Scenario C) with $R=0, \rho=0$ and $R=0, \rho\ne 0$. The dashed horizontal line marks the global optimal value ($y = 1747.41$), which was NOT achieved in the vast majority of runs, demonstrating the sensitivity of the algorithm to initial conditions.
• Figure 5: The data distribution of the Multivariate Cauchy Dataset.

Theorems & Definitions (17)

• Lemma 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Remark 6
• Theorem 7
• Theorem 8
• Example 1
• Theorem 9