In this paper we consider the numerical solution to the soft-margin support vector machine optimization problem. This problem is typically solved using the SMO algorithm, given the high computational complexity of traditional optimization algorithms when dealing with large-scale kernel matrices. In this work, we propose employing an NFFT-accelerated matrix-vector product using an ANOVA decomposition for the feature space that is used within an interior point method for the overall optimization problem. As this method requires the solution of a linear system of saddle point form we suggest a preconditioning approach that is based on low-rank approximations of the kernel matrix together with a Krylov subspace solver. We compare the accuracy of the ANOVA-based kernel with the default LIBSVM implementation. We investigate the performance of the different preconditioners as well as the accuracy of the ANOVA kernel on several large-scale datasets.

翻译：本文考虑软间隔支持向量机优化问题的数值求解。由于传统优化算法在处理大规模核矩阵时计算复杂度较高，该问题通常采用SMO算法进行求解。本文提出在整体优化问题的内点法中，利用基于ANOVA分解的特征空间，采用NFFT加速的矩阵向量乘积。由于该方法需要求解鞍点形式的线性方程组，我们提出一种基于核矩阵低秩近似的预处理方法，并结合Krylov子空间求解器。我们将基于ANOVA核的精度与默认LIBSVM实现进行了比较，并在多个大规模数据集上研究了不同预处理器的性能以及ANOVA核的精度。