Derivative of normalization function

Source: R/dA.R

This function computes the derivative of the function $$v\mapsto \frac{w}{\|w\|_A}$$ with respect to y.

dA(w, A, dw)

Arguments

w

vector of length n.
A

square matrix that defines the norm
dw

derivative of w with respect to y. As y is a vector of length n, the derivative is a matrix of size nxn.

Value

the Jacobian matrix of the normalization function. This is a matrix of size nxn.

Details

The first derivative of the normalization operator is $$\frac{\partial}{\partial y}\left(w\mapsto \frac{w}{\|w\|_A}\right)=\frac{1}{\|w\|}\left(I_n - \frac{w w^ \top A}{w^\top w}\right) \frac{\partial w}{\partial y}$$

References

Kraemer, N., Sugiyama M. (2011). "The Degrees of Freedom of Partial Least Squares Regression". Journal of the American Statistical Association 106 (494) https://www.tandfonline.com/doi/abs/10.1198/jasa.2011.tm10107

Kraemer, N., Braun, M.L. (2007) "Kernelizing PLS, Degrees of Freedom, and Efficient Model Selection", Proceedings of the 24th International Conference on Machine Learning, Omni Press, 441 - 448

See also

normalize, dnormalize

Author

Nicole Kraemer

Examples


w<-rnorm(15)
dw<-diag(15)
A<-diag(1:15)
d.object<-dA(w,A,dw)