Maximization problem of tuple set

[brydustin]

Given a set of vectors {v_ j } = {v_1, ... v_N} and I wish to transform each vector in the following manner

v_ i ' = Sum_k=1 to N ( c_ k,i) * v_i where c_ k,i is a scalar and what we are trying to solve for.

such that the sum of the distances squared between each pair of transformed vectors is maximized.

Then we would like to solve dD/dc = 0 where D is the sum of the distances squared and c are the scalars in matrix form. I'm not sure how to think of the c's except I could define a diagonal matrix whose entries are the sum of the respective scalars (i.e. i-th row is Sum_k=1 to N ( c_ k,i)) and multiply it by a vector (tensor) whose elements are the v_ j's so that the new vector has as its elements the transformed vectors.

 

[dan7]

We talked about it in chat --
He forgot to mention:

1. All vectors are in $$R^{3}$$
2. All v and v' are unit vectors (otherwise it's impossible to maximize)

So, I think first thing to do is find the rank of the Nx3 matrix of v's. If rank is < 3, the problem is trivial. Otherwise we just need to find a set of unit vectors that are 'equally' spread on a unit sphere. So, for N = 3 they would be coplanar with 120 degree angles, N=4 would make a tetrahedron, N = 6, 8 make cubes etc.

I don't know if there is a general closed formula for the N vectors. However, if we define energy functions $$\varphi_{ij} = - log(||x_{i} - x_{j}||^2)$$, and total energy function $$\Phi = \sum{\varphi_{ij}}$$, we can find partial derivatives of $$\Phi$$ wrt each $$x_{i}^{(k)}$$ and then use some algorithm like gradient descent to find the answer.

Danil