The covariance is defined as:
k(x, y) = sf^2 * exp(-1/2 * |(Q_i*P*(x-y))|^2)
where sf^2 is the amplitude, P a general d*D dimensionality reduction matrix (d << D), and Q_i is a d*d diagonal matrix containing all multiscales for inducing input number i.
Note that multiscales must not get smaller than 0.5 in this framework, because the overall length scale is considered to be equal to 1, which imposes this mathematical constraint for positive-definiteness. There is no need for a variable global length scale, because the dimensionality reduction matrix already generalizes this feature anyway. Hence an unconstrained multiscale parameter q is stored as log(q - 0.5).
If x and y are the same inducing input, then and only then extra noise (a different noise level for each inducing input) will be added for heteroskedasticity.
Dimensionality reduction, heteroskedasticity, and multiscales are optional features and can be easily turned off by setting the parameters to None.
module Params : sig ... endmodule Eval :
Interfaces.Specs.Eval
with type Kernel.params = Params.t
with type Inducing.t = Lacaml.D.mat
with type Input.t = Lacaml.D.vec
with type Inputs.t = Lacaml.D.matmodule Proj_hyper : sig ... endmodule Dim_hyper : sig ... endmodule Inducing_hyper : sig ... endmodule Hyper_repr : sig ... endmodule Deriv :
Interfaces.Specs.Deriv
with module Eval = Eval
with type Hyper.t = Hyper_repr.t