Delta Distancing: A Lifting Approach to Localizing Items from User Comparisons

Andrew D. McRae, Austin Xu, Jihui Jin, Namrata Nadagouda, Nauman Ahad, Peimeng Guan, Santhosh Karnik, Mark A. Davenport

Abstract: A common problem in recommendation systems is to learn a model of user preferences based only on comparisons of the relative attractiveness of different items. We consider this problem in the context of an ideal point model of user preference, where each user can be represented as a point in a low-dimensional space together with a set of items. In this model, the closer an item is to a user’s ideal point, the more that user prefers the item. When an embedding of items is known a priori, the problem of localizing a user’s ideal point from comparisons amongst items is well studied. However, relatively little work exists on learning embeddings for new items based only on such comparisons. In this paper, we consider the problem of embedding a set of items using paired comparisons from a set of known users. Specifically, we present a novel convex lifted method of learning the embedding representation p <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> ,…,p <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf> ∈ R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> of n items given noisy responses of the form "user u <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> prefers item p <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> to item p <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</inf> " for an arbitrary set of users {u <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</inf> } in R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> . We provide a range of simulations that validate the efficacy of our approach.