We prove a few representer theorems for a localised version of the regularised and multiview support vector machine learning problem introduced by H.Q. Minh, L. Bazzani, and V. Murino, Journal of Machine Learning Research, 17(2016) 1-72, that involves operator valued positive semidefinite kernels and their reproducing kernel Hilbert spaces. The results concern general cases when convex or nonconvex loss functions and finite or infinite dimensional input spaces are considered. We show that the general framework allows infinite dimensional input spaces and nonconvex loss functions for some special cases, in particular in case the loss functions are Gateaux differentiable. Detailed calculations are provided for the exponential least square loss function that lead to partially nonlinear equations for which a particular unconstrained potential reduction Newton's approximation method can be used.

翻译：我们证明了由H.Q. Minh、L. Bazzani和V. Murino在《机器学习研究杂志》第17卷（2016年）第1-72页提出的正则化与多视角支持向量机学习问题的局部化版本的若干表示定理，该问题涉及算子值半正定核及其再生核希尔伯特空间。这些结果适用于考虑凸或非凸损失函数以及有限或无限维输入空间的一般情形。我们证明该框架允许在某些特殊情况下使用无限维输入空间和非凸损失函数，特别是当损失函数为Gateaux可微时。针对指数最小二乘损失函数提供了详细计算，推导出可通过特定无约束势能下降牛顿逼近方法求解的部分非线性方程。