[R] fitting a curve according to a custom loss function

[M.Kondrin]

>
>
>Note that Vadim said he wanted to find f().  You're assuming f() is known.
>
>The model is very strange (to me, at least).  It's not obvious to me that
>it's even identifiable.  (Sorry that I don't have anything constructive to
>add.)
>
>Andy
> 
>  
>
Yes, I have missed the point.

But if you have a grid in (x_1,x_2) plane and y[i,j] values in the 
nodes  (or you can interpolate an irregularly spaced data) then you may 
solve your problem with Fourier transform, get fourier coefficients and 
find your function (i.e tabulated values) with inverse fourier transform.
Precisely it would look like that
Suppose there is a square grid in plane. Your model is 
y[i,j]=a0+a*(z<-cbind(c(f1,f2,f3...fn), c(f1,f2,f3,...fn)....))+t(z), 
i,j<N (because function f(x) is a table then may painlessly suppose that 
a2 is included in it). Then you convolve row-wise and column-wise y[i,j] 
with sin(2*pi*w*i/N), cos(...) (w<N). "Best-fit" Fourier coefficient 
(with multiplier "a" for columnwise convolution) is an average of vector 
you got after convolution.   a0 may be found then convolving with 1 
(i.e. just calculate average). After you got all coefficients "a" may be 
found with linear fit and (fi) with inverse fourier transform.
Fourier transform is little tricky when dealing with partial sums (with 
truncated series) because a result is not always smooth although in 
theory it should be, but why not give it a try.