SE!
Assume I have a function $f : \mathbb{R}^e \rightarrow \mathbb{R}^d$, is there a natural way to lift it to a function $f : \mathbb{R}^{e\times e} \rightarrow \mathbb{R}^{d \times d}$?
My concrete use is as follows, assume I have an operation $q$ that accepts two vectors $a : \mathbb{R}^d$ and $b : \mathbb{R}^d$ and produces a matrix $A : \mathbb{R}^{d \times d}$. This could be some outer product operation, e.g. $q(a, b) = \nabla_a f(a) \otimes \nabla_b g(b)$ (where $f, g: \mathbb{R}^d \rightarrow \mathbb{R}$). Now, I would like to specify $a$ and $b$ in terms of a deterministic invertible transform $T : \mathbb{R}^e \rightarrow \mathbb{R}^d$ from some other variables $x, y : \mathbb{R}^e$, so $a = T(x)$ and $b = T(y)$. The result of applying $q(T(x), T(y))$ would be in $\mathbb{R}^{d\times d}$, as expected, and I would like to convert the result to something in $R^{e\times e}$ using the inverse transform $T^{-1} : \mathbb{R}^d \rightarrow \mathbb{R}^e$.
Any idea how this would be possible?
Thanks a lot!