School of Information, Jiangnan University, Wuxi, Jiangsu, China.
Neural Netw. 2011 May;24(4):360-9. doi: 10.1016/j.neunet.2011.01.007. Epub 2011 Feb 3.
As we may know well, uniqueness of the Support Vector Machines (SVM) solution has been solved. However, whether Support Vector Data Description (SVDD), another best-known machine learning method, has a unique solution or not still remains unsolved. Due to the fact that the primal optimization of SVDD is not a convex programming problem, it is difficult for us to theoretically analyze the SVDD solution in an analogous way to SVM. In this paper, we concentrate on the theoretical analysis for the solution to the primal optimization problem of SVDD. We first reformulate equivalently the primal optimization problem of SVDD into a convex programming problem, and then prove that the optimal solution with respect to the sphere center is unique, derive the necessary and sufficient conditions of non-uniqueness of the optimal solution with respect to the sphere radius in the primal optimization problem of SVDD. Moreover, we also explore the property of the SVDD solution from the perspective of the SVDD dual form. Furthermore, according to the geometric interpretation of SVDD, a method of computing the sphere radius is proposed when the optimal solution with respect to the sphere radius in the primal optimization problem is non-unique. Finally, we have several examples to illustrate these findings.
众所周知，支持向量机（SVM）的解具有独特性。然而，另一种广为人知的机器学习方法——支持向量数据描述（SVDD）是否具有独特的解，尚未得到解决。由于 SVDD 的原始优化问题不是凸规划问题，因此我们很难以类似于 SVM 的方式从理论上分析 SVDD 解。在本文中，我们专注于对 SVDD 原始优化问题解的理论分析。我们首先将 SVDD 的原始优化问题等效地重新表述为凸规划问题，然后证明了关于球心的最优解是唯一的，推导出了 SVDD 原始优化问题中关于球半径的最优解非唯一性的充要条件。此外，我们还从 SVDD 对偶形式的角度探讨了 SVDD 解的性质。此外，根据 SVDD 的几何解释，当原始优化问题中关于球半径的最优解不唯一时，提出了一种计算球半径的方法。最后，我们通过几个例子来说明这些发现。
