Asymmetric data naturally exist in real life, such as directed graphs. Different from the common kernel methods requiring Mercer kernels, this paper tackles the asymmetric kernel-based learning problem. We describe a nonlinear extension of the matrix Singular Value Decomposition through asymmetric kernels, namely KSVD. First, we construct two nonlinear feature mappings w.r.t. rows and columns of the given data matrix. The proposed optimization problem maximizes the variance of each mapping projected onto the subspace spanned by the other, subject to a mutual orthogonality constraint. Through Lagrangian duality, we show that it can be solved by the left and right singular vectors in the feature space induced by the asymmetric kernel. Moreover, we start from the integral equations with a pair of adjoint eigenfunctions corresponding to the singular vectors on an asymmetrical kernel, and extend the Nystr\"om method to asymmetric cases through the finite sample approximation, which can be applied to speedup the training in KSVD. Experiments show that asymmetric KSVD learns features outperforming Mercer-kernel based methods that resort to symmetrization, and also verify the effectiveness of the asymmetric Nystr\"om method.

翻译：非对称数据在现实生活中普遍存在，例如有向图。与常见的基于Mercer核的核方法不同，本文探讨了基于非对称核的学习问题。我们通过非对称核描述了矩阵奇异值分解的非线性扩展，即KSVD。首先，我们针对给定数据矩阵的行和列分别构建两种非线性特征映射。所提出的优化问题在满足相互正交约束的条件下，最大化每个映射投影到另一个映射张成子空间上的方差。通过拉格朗日对偶性，我们证明该问题可通过非对称核诱导特征空间中的左右奇异向量求解。此外，我们从非对称核上对应奇异向量的一对伴随本征函数的积分方程出发，通过有限样本近似将Nyström方法扩展到非对称情形，从而加速KSVD的训练。实验表明，非对称KSVD学习到的特征优于依赖对称化处理的基于Mercer核的方法，同时也验证了非对称Nyström方法的有效性。