I have this ill-posed objective function $\min_{z}f(z)=\|1-k(z)\|$, where $z$ is two dimensional points such that $z_2+\sigma=z_1^2+\sigma$. Based on the assumption that, the underlying manifold of $z$ is one of the level set of $f(z)$. I am approximate this contour by find the step that the first stationary point at along opposite gradient direction at that point $(z,f(z))$, and then apply the same step and direction to the 2D point $z$. The result is in the middle layer in the figure 1. Here is a sample image. The top surface is a user-defined l2 loss function over points in middle. The thick curve at the bottom is my current approximation results.
I am looking for an approximation of 2D manifold(which should be a parabola). Can anyone recommend some researches focusing on the problem that finding the best-fit level sets against feasible sets in terms of some norm.