Learning Positive Functions in a Hilbert Space

We develop a method for learning positive functions by optimizing over SoSK, a reproducing kernel Hilbert space subject to a Sum-of-Squares (SoS) constraint. This constraint ensures that only nonnegative functions are learned. We establish a new representer theorem that demonstrates that the regularized convex loss minimization subject to the SoS constraint has a unique solution and moreover, its solution lies on a finite dimensional subspace of an RKHS that is defined by data. Furthermore, we show how this optimization problem can be formulated as a semidefinite program. We conclude with an example of learning such functions.