Privacy-preserving linear and nonlinear approximation via linear programming

We propose a novel privacy-preserving random kernel approximation based on a data matrix A ∈ R whose rows are divided into privately owned blocks. Each block of rows belongs to a different entity that is unwilling to share its rows or make them public. We wish to obtain an accurate function approximation for a given y ∈ Rm corresponding to each of the m rows of A. Our approximation of y is a real function on Rn evaluated at each row of A and is based on the concept of a reduced kernel K(A,B) where B is the transpose of a completely random matrix B. The proposed linear-programming-based approximation, which is public but does not reveal the privately-held data matrix A, has accuracy comparable to that of an ordinary kernel approximation based on a publicly disclosed data matrix A.