Dispersiekernels

Dispersiekernels are kernel functions devised to capture dispersion, or spread, in data by allowing the scale of similarity to vary with location. They are used when data exhibit nonstationary dispersion, meaning that the typical distance between similar observations changes across the input space. By incorporating a dispersion function, dispersiekernels generalize standard kernels to adapt to local density or variability.

A common construction defines a local bandwidth function σ(x) > 0 and sets k_disp(x, y) = exp(- ||x - y||^2 / (2 (σ(x)^2 + σ(y)^2))). This yields greater or lesser similarity depending on the local dispersion around x and y. More generally, a locally varying metric M(x) (a positive definite matrix) can be used, giving k_disp(x, y) = exp(- (x - y)^T [M(x) + M(y)] (x - y)/2). In either form, the kernel reduces to a standard stationary kernel like the Gaussian when the dispersion is constant across the input space.

Properties and caveats: symmetry k_disp(x, y) = k_disp(y, x) typically holds, but positive definiteness is not guaranteed when dispersion varies with location. Practitioners may need to enforce PD by construction or use approximate inference methods that tolerate non-PD kernels.

Applications are in nonparametric regression and density estimation with adaptive bandwidth, Gaussian processes that model heteroscedastic noise or nonstationarity, spectral clustering on data with varying dispersion, and manifold learning where local spread changes across regions.

See also: kernel methods, adaptive bandwidth, kernel density estimation, Gaussian processes, diffusion kernels, heteroscedastic modeling.