This paper investigates Support Vector Regression (SVR) in the context of the fundamental risk quadrangle theory, which links optimization, risk management, and statistical estimation. It is shown that both formulations of SVR, $\varepsilon$-SVR and $\nu$-SVR, correspond to the minimization of equivalent error measures (Vapnik error and CVaR norm, respectively) with a regularization penalty. These error measures, in turn, define the corresponding risk quadrangles. By constructing the fundamental risk quadrangle, which corresponds to SVR, we show that SVR is the asymptotically unbiased estimator of the average of two symmetric conditional quantiles. Further, we prove the equivalence of the $\varepsilon$-SVR and $\nu$-SVR in a general stochastic setting. Additionally, SVR is formulated as a regular deviation minimization problem with a regularization penalty. Finally, the dual formulation of SVR in the risk quadrangle framework is derived.

翻译：本文在基础风险四边形理论的背景下研究了支持向量回归（SVR），该理论将优化、风险管理和统计估计联系起来。研究表明，SVR的两种形式，即$\varepsilon$-SVR和$\nu$-SVR，分别对应于带有正则化惩罚的等价误差度量（Vapnik误差和CVaR范数）的最小化。这些误差度量进而定义了相应的风险四边形。通过构建与SVR相对应的基础风险四边形，我们证明SVR是两个对称条件分位数平均值的渐近无偏估计量。此外，我们在一股随机环境下证明了$\varepsilon$-SVR和$\nu$-SVR的等价性。进一步地，SVR被表述为带有正则化惩罚的正则偏差最小化问题。最后，推导了风险四边形框架下SVR的对偶形式。