Nyström Kernel Mean Embeddings
This work addresses scalability issues for researchers and practitioners using kernel methods in machine learning, though it is incremental as it builds on existing Nyström techniques.
The paper tackles the computational and storage inefficiency of kernel mean embeddings in large-scale settings by proposing a Nyström-based approximation using random subsets, achieving a standard n^{-1/2} error rate with reduced costs.
Kernel mean embeddings are a powerful tool to represent probability distributions over arbitrary spaces as single points in a Hilbert space. Yet, the cost of computing and storing such embeddings prohibits their direct use in large-scale settings. We propose an efficient approximation procedure based on the Nyström method, which exploits a small random subset of the dataset. Our main result is an upper bound on the approximation error of this procedure. It yields sufficient conditions on the subsample size to obtain the standard $n^{-1/2}$ rate while reducing computational costs. We discuss applications of this result for the approximation of the maximum mean discrepancy and quadrature rules, and illustrate our theoretical findings with numerical experiments.