Side information is highly useful in the learning of a nonparametric kernel matrix. However, this often leads to an expensive semidefinite program (SDP). In recent years, a number of dedicated solvers have been proposed. Though much better than off-the-shelf SDP solvers, they still cannot scale to large data sets. In this paper, we propose a novel solver based on the alternating direction method of multipliers (ADMM). The key idea is to use a low-rank decomposition of the kernel matrix $\K = \V^\top \U$, with the constraint that $\V=\U$. The resultant optimization problem, though non-convex, has favorable convergence properties and can be efficiently solved without requiring eigen-decomposition in each iteration. Experimental results on a number of real-world data sets demonstrate that the proposed method is as accurate as directly solving the SDP, but can be one to two orders of magnitude faster.

翻译：边信息在非参数核矩阵学习中非常有用，但往往会导致求解昂贵的半定规划问题。近年来，研究人员提出了多种专用求解器。尽管这些求解器远优于现成的半定规划求解器，但仍无法扩展到大规模数据集。本文提出一种基于交替方向乘子法的新颖求解器，其核心思想是采用核矩阵的低秩分解 $\K = \V^\top \U$，并施加约束 $\V=\U$。所得优化问题虽为非凸，但具有良好的收敛性质，且无需在每次迭代中执行特征分解即可高效求解。多个真实数据集上的实验结果表明，该方法在保持与直接求解半定规划同等精度的同时，可实现一至两个数量级的加速。