Separable function approximation (minimax or least square)

Question

I have no idea if this is a thing but I want to approximate a function $f(x,y)$ by $g(x)h(y)$

$$\underset{g,h}{\text{argmin}} {\iint}\bigg(f(x,y)-g(x)h(y)\bigg)^2\,dxdy$$

Is there any numerical or analytical approach to this?

Answer 1

If you sample the values of $f$ into a matrix $$F = \begin{bmatrix}f(x_1,y_1) & f(x_1,y_2) & \cdots \\ f(x_2,y_1) & f(x_2,y_2) & \cdots \\ \vdots & \vdots & \ddots\end{bmatrix},$$ then your problem is equivalent to finding the best rank-$1$ approximation $$F \approx \begin{bmatrix}g(x_1) \\ g(x_2) \\ \vdots\end{bmatrix}\begin{bmatrix}h(y_1) & h(y_2) & \cdots\end{bmatrix}$$ under the Frobenius norm. This can be solved by computing the SVD and retaining only the first singular value.