Localizing users and items from paired comparisons

Suppose that we wish to determine an embedding of points given only paired comparisons of the form “user x prefers item qi to item qj.” Such observations arise in a variety of contexts, including applications such as recommendation systems, targeted advertisement, and psychological studies. In this paper we first present an optimization-based framework for localizing new users and items when an existing embedding is known. We demonstrate that user localization can be formulated as a simple constrained quadratic program, and that although item localization produces a set of non-convex constraints, we can make the problem convex by strategically combining comparisons to produce a set of convex linear constraints. Finally, we show that by iteratively applying this method to every user and item, we can recover an accurate embedding, allowing us to iteratively improve a given embedding or even generate an embedding from scratch.