NKF0

NKF0 is a theoretical kernel construct in the field of kernel-based data analysis. It denotes a centered, zero-mean variant of a base kernel K, designed to produce feature maps with vanishing mean when data are drawn from the underlying distribution. This centering mirrors similar normalization steps used in linear methods and is discussed in the context of reproducing kernel Hilbert spaces.

Definition and construction: The NKF0 kernel K_c is defined by K_c(x, x') = K(x, x') - E_x[K(x, x')] - E_x'[K(x, x')] + E[K(X, X')], where the expectations are taken with respect to the data distribution P (and X, X' are independent draws from P). In practice, if P is unknown, empirical centering uses the training sample to estimate the expectations, resulting in an empirical NKF0 kernel K_c_emp.

Properties: K_c is symmetric and positive semidefinite if K is PSD. The corresponding feature map φ_c satisfies E_{x~P}[φ_c(x)] = 0. NKF0 preserves inner products after centering, so kernel methods relying on variance structure remain applicable while first-order mean differences are removed.

Applications and considerations: NKF0 is used to stabilize kernel-based algorithms such as kernel PCA, kernel ridge regression, and clustering in the presence of strong mean structure. It aids in domain adaptation by reducing mean mismatches between domains. Practical use requires accurate estimation of the centering terms; for large datasets, tricks such as block-wise or iterative centering can reduce computation.

See also: kernel methods, reproducing kernel Hilbert space, centered kernel, kernel PCA.