Irregular Sampling of High-Dimensional Functions in Reproducing Kernel Hilbert Spaces
This work addresses computational efficiency in high-dimensional function approximation for researchers in numerical analysis and machine learning, but appears incremental as it builds on existing tensor product kernel methods.
The paper tackles the problem of sampling high-dimensional functions in reproducing kernel Hilbert spaces using irregular samples, and shows that composing lower-dimensional irregular samples into tensors significantly reduces computational complexity.
We develop sampling formulas for high-dimensional functions in reproducing kernel Hilbert spaces, where we rely on irregular samples that are taken at determining sequences of data points. We place particular emphasis on sampling formulas for tensor product kernels, where we show that determining irregular samples in lower dimensions can be composed to obtain a tensor of determining irregular samples in higher dimensions. This in turn reduces the computational complexity of sampling formulas for high-dimensional functions quite significantly.