# Estimating the dimension of a solution space in nonlinear least squares

Suppose I have a nonlinear least squares problem, $$\min_{\mathbf{x}} || \mathbf{f}(\mathbf{x}) ||^2$$ with $$n$$ residuals and $$m$$ parameters, so that $$\mathbf{x} \in \mathbb{R}^m$$, and $$\mathbf{f} \in \mathbb{R}^n$$, with $$n > m$$. Suppose there are multiple global minima $$\mathbf{x}$$ which minimize the function above. I would like to estimate how large this solution space is (i.e. roughly how many different $$\mathbf{x}$$ vectors are global minima). Is there an algorithm which will help with this?

I assume it would need to be a Monte-Carlo type method, picking lots of random initial conditions, then running the NLS solver to get to the solution, and then counting the number of distinct solutions found. But if anyone knows of any relevant publications for this type of problem, please tell me.

• I suspect that any meaningful estimates would depend enormously on some pretty strict regularity assumptions on $f$; i.e., coercivity, smoothness, good behavior at fine scales, etc. I'm sure there are some results in this direction but global optimization is perhaps the epitome of "no free lunch" so I doubt there are any methods that are both general and useful – whpowell96 Jun 21 at 18:24