Derivative of normalization function

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

Usage

dnormalize(v, dv)

Arguments

v: vector of length n.
dv: derivative of v 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 $$\frac{\partial}{\partial y}\left(v\mapsto \frac{v}{\|v\|}\right)=\frac{1}{\|v\|}\left(I_n - \frac{v v^ \top}{v^\top v}\right) \frac{\partial v}{\partial y}$$

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

Author

Nicole Kraemer, Mikio L. Braun

Examples


v<-rnorm(15)
dv<-diag(15)
d.object<-dnormalize(v,dv)