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

Nonlinear SVD with Asymmetric Kernels: feature learning and asymmetric Nystr\"om method