Derivative of normalization function
This function computes the derivative of the function
v\mapsto \frac{v}{\|v\|}
with respect to y.
dnormalize(v, dv)
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. |
The first derivative of the normalization operator is
\frac{\partial}{\partial y}≤ft(v\mapsto \frac{v}{\|v\|}\right)=\frac{1}{\|v\|}≤ft(I_n - \frac{v v^ \top}{v^\top v}\right) \frac{\partial v}{\partial y}
the Jacobian matrix of the normalization function. This is a matrix of size nxn.
Nicole Kraemer, Mikio L. Braun
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
v<-rnorm(15) dv<-diag(15) d.object<-dnormalize(v,dv)
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