Mercer's theorem

Mercer's theoremone ofmost popular results ofworkJames Mercer. Itused forkernel trick, an important methodmachine learning.

Mercer's theorem states that any positive definite kernel K(x, y) can be expressed asdot product inhigh-dimensional space.

More specifically, ifkernelpositive semi-definite, i.e.,

then there existsfunction whose imagein an inner product spacepossibly high dimension, such that