Learning in Feature Spaces via Coupled Covariances: Asymmetric Kernel SVD and Nyström method
This work addresses a foundational problem in kernel methods for machine learning, offering incremental improvements to asymmetric kernel techniques.
The authors tackled the limitations of existing Asymmetric Kernel SVD (KSVD) by introducing a new paradigm based on coupled covariance eigenproblems, enabling infinite-dimensional feature maps and formalizing an asymmetric Nyström method for faster training, with empirical evaluations showing practical utility across multiple tasks.
In contrast with Mercer kernel-based approaches as used e.g., in Kernel Principal Component Analysis (KPCA), it was previously shown that Singular Value Decomposition (SVD) inherently relates to asymmetric kernels and Asymmetric Kernel Singular Value Decomposition (KSVD) has been proposed. However, the existing formulation to KSVD cannot work with infinite-dimensional feature mappings, the variational objective can be unbounded, and needs further numerical evaluation and exploration towards machine learning. In this work, i) we introduce a new asymmetric learning paradigm based on coupled covariance eigenproblem (CCE) through covariance operators, allowing infinite-dimensional feature maps. The solution to CCE is ultimately obtained from the SVD of the induced asymmetric kernel matrix, providing links to KSVD. ii) Starting from the integral equations corresponding to a pair of coupled adjoint eigenfunctions, we formalize the asymmetric Nyström method through a finite sample approximation to speed up training. iii) We provide the first empirical evaluations verifying the practical utility and benefits of KSVD and compare with methods resorting to symmetrization or linear SVD across multiple tasks.