Estimate Conditional Mean Embedding via Kernel Ridge Regression

Estimates the conditional mean embedding \(\mu_{Y|X=x}\) in the RKHS using kernel ridge regression (Park, Muandet, Fukumizu & Sejdinovic, 2013; Muandet et al., 2017).

Usage

fit_cme(
  x,
  y,
  kernel_x = kernel_spec(),
  kernel_y = kernel_spec(),
  lambda = "cv"
)

Arguments

x

Numeric matrix of conditioning variables (n x d_x).

y

Numeric matrix of target variables (n x d_y).

kernel_x

Kernel specification for x.

kernel_y

Kernel specification for y.

lambda

Ridge regularisation parameter. If "cv", selected by leave-one-out cross-validation. Default is "cv".

Value

A list of class "cme_fit" with components:

W

Operator matrix (K_x + n lambda I)^{-1} (n x n).

Ky

Kernel matrix of y.

x_train

Training x data.

kernel_x, kernel_y

Resolved kernel specifications.

lambda

Regularisation parameter used.

Details

This is the lower-level building block used by kernel_downscale(). Most users should call kernel_downscale(); use fit_cme() directly when you need access to the trained operator (the weight matrix W) for custom downstream computations – e.g. constructing a kernel Bayes' rule update, plugging into a manuscript figure pipeline, or composing with other RKHS operators.

References

Park, J., Muandet, K., Fukumizu, K., & Sejdinovic, D. (2013). Kernel embeddings of conditional distributions: A unified kernel framework for nonparametric inference in graphical models. IEEE Signal Processing Magazine.

Muandet, K., Fukumizu, K., Sriperumbudur, B., & Scholkopf, B. (2017). Kernel mean embedding of distributions: A review and beyond. Foundations and Trends in Machine Learning, 10(1-2).

See also

predict.cme_fit(), kernel_downscale()

Other downscaling and embeddings: aggregate_downscale(), dist_regression(), kernel_downscale(), posterior_sample_aggregate()

Examples

set.seed(1L)
x <- matrix(rnorm(60L), ncol = 2L)
y <- matrix(x[, 1L] + rnorm(30L, sd = 0.2), ncol = 1L)
fit <- fit_cme(x, y, lambda = 1e-2)
dim(fit$W)
#> [1] 30 30