That doesn't quite work out that way I think, if you compare it to solving a system of equations then the size of the input is irrelevant. Indeed a very large i

That doesn't quite work out that way I think, if you compare it to solving a system of equations then the size of the input is irrelevant. Indeed a very large input is often the main reason for a problem to be under-specified. What you should look at is the number of outputs times the number of data points for each output. If this number is lower than the number of parameters then it should be possible to find multiple solutions. Of course in this case you're not looking for a solution, but an optimum, and not even a global one, so it's not too troubling per se that you don't get a unique answer. Though it does somewhat suggest you should be able to get an equivalent fit with far fewer parameters, but finding it could be quite tricky.