Split matrix so that new matrices have equivalent properties

Hello,

I have a large 400x2 matrix.

There are 400 items, which have the properties A and B (correspond to the columns).

The goal is to decompose the matrix into 10 matrices of 40 items each, so that in the end I have 40 10x2 matrices.

The matrices should be comparable regarding their mean values per column. Therefore, there is probably an optimization problem here.

The target is that column 1 of the matrices roughly corresponds to the mean value of A and column 2 roughly corresponds to the mean value of B.

In the first step, I randomly decomposed the large matrix into 40 10x2 matrices. This represents the initial position.

I know that I have to swap the rows between the matrices until the target is reached, that is, I have reached the minima (difference between target value and current mean). However, I do not know how to solve this most effectively.

I am looking forward to any tips and suggestions.

Thank you in advance :-)

Answer 1

[Walter Roberson]

If you only had one value to worry about, then there would be an algorithm:

after you have randomly split the matrix, repeat:

swap the smallest row of the matrix with the smallest mean, with the largest row of the matrix with the largest mean

At the moment I am not sure what the termination condition would be.

However with you having two values to worry about, the general case might be much harder. I am not sure at the moment whether it is always possible; I worry that there might be constructed conditions that cannot be improved, since there is an implication that a linear ordering might have to be imposed on 2d data.

[Sasse17]

Yes, one dimension would be easier...

I was thinking that for the first column it is possible to do as you mentioned. For the second column you do the same, but you have to check that the value of column 1 is almost the same.

But the implementation is a problem. Is there already code for this? I don't know what else to look for....