Least squares with multi-dimensional output type

Specific Domains Optimization (Mathematical)
1

I am trying to use LsqFit to find out a 2D geometric transformation parameter, say a mapping from (x,y) to (xp,yp):

x = [x for x=-2.:2. for y=-2.:2.]
y = [y for x=-2.:2. for y=-2.:2.]
in_data = [x y]
k = -0.02
@. xp = x + k*x*(x^2+y^2)
@. yp = y + k*y*(x^2+y^2)
out_data = [xp yp]

However, from the readme of LsqFit it’s not clear how to accommodate multi-dimensional dependent variable (or is it possible?). I tried to format the dependent variable in the same format as the independent variable, which is a nx2 matrix, as follows:

k0=[0.]
function model(x,p) 
    @. xp = x[:,1]+p[1]*x[:,1]*(x[:,1]^2+x[:,2]^2) 
    @. yp = x[:,2]+p[1]*x[:,2]*(x[:,1]^2+x[:,2]^2)
    return [xp yp]
end
ret = curve_fit(model, in_data, out_data, k0)

But this doesn’t work and I got the error message:

LoadError: MethodError: no method matching mul!(::Array{Float64,1}, ::LinearAlgebra.Transpose{Float64,Array{Float64,2}}, ::Array{Float64,2}, ::Bool, ::Bool)

I saw this post which addresses the multivariate input variable, but I still can’t figure out how to deal with the output variable.
This example can actually be solved analytically, but I just wanted to find out how to do least squares for 2D points.
Originally posted here.

2

Please link to your SO question if you cross-post in multiple places, the question was asked here: julia - Use LsqFit for multi-variate output? - Stack Overflow

As I said in the comment there, I don’t see how you find a minimum here without defining some sort of norm that quantifies the distance between candidate solutions? Check out the tutorial in the LsqFit docs to see how the package estimates parameters - basically by minimzing the sum of squared distances between your output data and f(input data, parameters).

1 Like
3

Added the link to the original post.
I think conceptually the loss function is simply the sum of Euclidean distances between points. It should be quite straight forward to compute. Maybe LsqFit didn’t implement higher dimensional dependent variables?

4

Going to answer my own question:
The vector output variable needs to be stacked together to form an 1D array. So the only changes needed is:

out_data = [xp; yp]