For reference we have the taylor series expansion of a vector:
$\textbf x = \left[\begin{array}{c}{x_1} \\ {x_2}\end{array}\right]$
$\nabla f = \left[\begin{array}{c}f_{x_1} & f_{x_2}\end{array}\right]$
$H = \left[\begin{array}{cc}f_{x_{1}x_{1}} & f_{x_{1}x_{2}}\\ f_{x_{1}x_{2}} & f_{x_{2}x_{2}}\end{array}\right]$
$f(\textbf x) = f(\textbf x^{0}) + (\textbf x-\textbf x^0)^T\nabla_\textbf x f(\textbf x^{0}) + \frac{1}{2!}(\textbf x-\textbf x^0)^TH(\textbf x-\textbf x^0) + ...$
Instead of expanding by looking at $x_1, x_2, x_n$ as above it would be convenient to redefine the coordinate system so to expand in terms of the unit vectors of $\textbf x_1, \textbf x_2, \textbf x_m$. (The reason being that $m<<n$). $f$ has the properties of a norm.
new unit vector, $\vec {\textbf x_1} = \frac{\textbf x_1}{f(\textbf x_1)}$
new vector, $\textbf X = \left[\begin{array}{c}{\frac{\textbf x_1}{f(\textbf x_1)}} \\ \frac{\textbf x_2}{f(\textbf x_2)}\end{array}\right]$
I would like to expand the function $f(\textbf X)$ and express it in terms of the unit vectors $\vec {\textbf x_1}, \vec {\textbf x_2}$.
Maybe this is more a dimensionality reduction problem but I am not happy with my definitions of $\textbf X$ and $\vec {\textbf x_1}$. I keep going in circles when I think about it. Does what I'm saying above make sense? I would be very grateful if someone could put me on the right path here or explain where I am going wrong.