Help finding matrix terms using Optimization

[Andrew Gibb]

Hi,

I'm a novice when it comes to optimization, so please forgive mis-use of terms, etc. I'll be as clear as I can.

I have a data set of 128-term vectors, which I suspect are related to a corresponding set of 128-term vectors, by a matrix transform. That is, for each vector d there is a corresponding vector dhat = Bd, where B is a 128x128 matrix. I can generate very many of these vector pairs. Each term in the vectors is an 8 bit integer.

I suspect that I can find the matrix B using some numerical optimisation, using the vector pairs as input, but I don't know where to start. I read the documentation on JuMP, but I couldn't understand how to specify the data. I guess the terms in B are a variable, but that's as far as I got.

Bonus 1: There is likely to be some degree of sparsity in B. The data can probably be structured such that the most important terms are expected near the diagonal, and terms farther from the diagonal are less relevant. 

Bonus 2: Each pair of vectors could have associated with it a set of four parameters, which could vary, and which the terms of B might be functions of. I'd like to understand how to solve the simpler case above first, though. 

Regards

Andy