Geometric Learning with Positively Decomposable Kernels
This work provides a theoretical foundation for kernel methods on non-Euclidean data, addressing a bottleneck in geometric learning, though it appears incremental as it builds on existing RKKS frameworks.
The paper tackles the difficulty of constructing positive-definite kernels for non-Euclidean data spaces by proposing the use of reproducing kernel Krein space (RKKS) methods, which only require positively decomposable kernels, and shows that invariant kernels admit such decompositions under tractable conditions, making kernel learning more accessible for non-Euclidean data.
Kernel methods are powerful tools in machine learning. Classical kernel methods are based on positive-definite kernels, which map data spaces into reproducing kernel Hilbert spaces (RKHS). For non-Euclidean data spaces, positive-definite kernels are difficult to come by. In this case, we propose the use of reproducing kernel Krein space (RKKS) based methods, which require only kernels that admit a positive decomposition. We show that one does not need to access this decomposition in order to learn in RKKS. We then investigate the conditions under which a kernel is positively decomposable. We show that invariant kernels admit a positive decomposition on homogeneous spaces under tractable regularity assumptions. This makes them much easier to construct than positive-definite kernels, providing a route for learning with kernels for non-Euclidean data. By the same token, this provides theoretical foundations for RKKS-based methods in general.