Localisation of Regularised and Multiview Support Vector Machine Learning
This work provides theoretical extensions for support vector machines, but appears incremental as it builds directly on prior research.
The authors developed a localized version of regularized and multiview support vector machines, proving representer theorems for cases with convex or nonconvex loss functions and finite or infinite dimensional input spaces. They demonstrated tractability through numerical experiments on a toy model.
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 squares loss functions that lead to systems of partially nonlinear equations for which a particular different types of Newton's approximation methods based on the interior point method can be used. Some numerical experiments are performed on a toy model that illustrate the tractability of the methods that we propose.