Scalable Learning in Reproducing Kernel Krein Spaces
This work addresses the challenge for practitioners in domains like graphs and time-series who use intuitive similarity functions that are often indefinite, enabling scalable learning without requiring positive definiteness verification.
The authors tackled the problem of large-scale learning with indefinite kernels by providing the first complete derivation of the Nyström method for such kernels and proposing efficient eigendecomposition techniques, resulting in scalable methods for reproducing kernel Krein spaces.
We provide the first mathematically complete derivation of the Nyström method for low-rank approximation of indefinite kernels and propose an efficient method for finding an approximate eigendecomposition of such kernel matrices. Building on this result, we devise highly scalable methods for learning in reproducing kernel Kreĭn spaces. The devised approaches provide a principled and theoretically well-founded means to tackle large scale learning problems with indefinite kernels. The main motivation for our work comes from problems with structured representations (e.g., graphs, strings, time-series), where it is relatively easy to devise a pairwise (dis)similarity function based on intuition and/or knowledge of domain experts. Such functions are typically not positive definite and it is often well beyond the expertise of practitioners to verify this condition. The effectiveness of the devised approaches is evaluated empirically using indefinite kernels defined on structured and vectorial data representations.