regression on simplexes

[Nickolay Trendafilov]

Hi, my field of interests is multivariate data analysis. Nevertheless, I have a problem that might be connected to your experience, or at least some ideas might come up. 

Let V_p be a standard p-dimensional simplex. We are given n pairs of data (x_i, y_i), i=1,...,n, such that x_i is from the interior of V_p and y_i is from the interior of V_q (p>q). The purpose is to find a linear (in some sense) map f, that connects x_i's with y_i's in the best possible way, say, in least squares sense, i.e. to minimize \sum_{I=1}^n ||y_i - f(x_i)||^2 or \sum_{I=1}^n ||log(y_i) - log(f(x_i))||^2.  

Data on a simplex V_p are known as compositional data. The summation of compositions and multiplication by a scalar is defined by Aitchison, and they make V_p an Euclidean vector space.

Thanks in advance for any suggestions, 

Nickolay