I have a map that projects a $k$ dimensional vector $x$ to an $m$ dimensional vector $\phi(x)$.
The vector function (map) $\phi$ can be any linear or non-linear function of $x$, which is not necessarily invertible.
I define: $$<x,y>:=\phi(x)^T\phi(y)=\sum_{i=1}^m \phi_i(x)\phi_i(y)$$
Now I'm looking for an estimated $\theta$, such that:
$$<x,\theta>:=\phi(x)^T\phi(\theta)=\sum_{i=1}^m \phi_i(x)*1=\sum_{i=1}^m \phi_i(x)$$
My try: Clearly $$<x,\theta>:=\phi(x)^T\phi(\theta)=\sum_{i=1}^m \phi_i(x)*1=\sum_{i=1}^m \phi_i(x)=\phi(x)^T\mathbf{1}$$, so we are looking for $\theta$'s that satisfy $\phi(\theta)=\mathbf{1}$, or equivalently: $\phi^{-1}(\mathbf{1})=\theta$.
I'm stuck here, because $\phi$ can be an arbitrary --highly nonlinear-- map, which is not necessarily invertible.
I think there should be a solution for efficiently approximating $\theta$.
My question: I don't need the set of $\theta$'s that exactly satisfy this constraint. how can I estimate $\theta$ from sample projections?
Thanks!