Pseudo inputs for pairwise learning with Gaussian processes

We consider learning and prediction of pairwise comparisons between instances. The problem is motivated from a perceptual view point, where pairwise comparisons serve as an effective and extensively used paradigm. A state-of-the-art method for modeling pairwise data in high dimensional domains is based on a classical pairwise probit likelihood imposed with a Gaussian process prior. While extremely flexible, this non-parametric method struggles with an inconvenient O(n3) scaling in terms of the n input instances which limits the method only to smaller problems. To overcome this, we derive a specific sparse extension of the classical pairwise likelihood using the pseudo-input formulation. The behavior of the proposed extension is demonstrated on a toy example and on two real-world data sets which outlines the potential gain and pitfalls of the approach. Finally, we discuss the relation to other similar approximations that have been applied in standard Gaussian process regression and classification problems such as FI(T)C and PI(T)C.