For a given set of data points

$$\{(x_i, y_i, z_i)\}$$

there exists some


that minimizes


$A$, $B$, and $C$ can be found quickly by directly solving/minimizing


However, suppose we want to reduce the model's degrees of freedom from 3 to 2 by constraining $f_{ABC}$ to be a parabolic sheet. I think this would mean that $\begin{bmatrix}A&B&C\end{bmatrix}$ has to be of the form $\begin{bmatrix}U^2&2UV&V^2\end{bmatrix}$ so that $f_{ABC}(x,y)=(Ux+Vy)^2$.

It seems this is no longer solvable directly as a linear system.

What would be a performant/fast way to solve this constrained version of the problem?


The above question is a simpler version of my actual problem, which is to fit an explicit bicubic surface patch $g(x,y)=Ax^2+Bxy+Cy^2+D+Ex^3+Fy^3+Gx^2y+Hxy^2+Ix+Jy$ that is constrained to be quadratic (instead of cubic) along some (any) direction.


Thinking further, my best idea currently is to

  1. Directly fit an unconstrained quadratic surface i.e. find a solution in the span of $\begin{bmatrix}x_i^2&x_iy_i&y_i^2&1&x_i&y_i\end{bmatrix}$. Call this the "quadratic solution".
  2. Orthogonalize the remaining basis vectors $\begin{bmatrix}x_i^3&y_i^3&x_i^2y_i&x_iy_i^2\end{bmatrix}$ w.r.t. the quadratic solution.
  3. Somehow solve a constrained minimization within this reduced basis that is orthogonal to the quadratic solution, though I don't really know how to go about this.

I think the eventual cubic terms should be of the form $Ex^3+Fy^3+Gx^2y+Hxy^2=(Ux+Vy)^3$ so the constraint is $\begin{bmatrix}E&F&G&H\end{bmatrix}=\begin{bmatrix}U^3&3U^2V&3UV^2&V^3\end{bmatrix}$

How to do this constrained optimization?


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