Matrix Factorization for Collaborative Budget Allocation

Abstract

This paper studies the collaborative budget allocation problem in which users are not isolated in the collaborative consumption of goods or services when available goods or services are limited. Different from existing methods that treat each user independently, we investigate the geometric properties of user’s consumption or preference on services, and design a matrix completion framework on the simplex. In this framework, an item’s allocation vector indicating how available services are allocated to users is estimated by the combination of user profiles as basis points on the simplex. Instead of using Euclidean distance directly, we specify a Riemannian distance on the simplex or project histogram data on simplex to Euclidean space. To intensify our model’s stability, we relax the exact recovery constraint to make a robust collaborative prediction. The resulting objective function is then efficiently optimized by a Riemannian conjugate gradient method on the simplex. Experiments on real-world data sets demonstrate our model’s competitiveness versus other collaborative budget prediction methods. Comparisons of different distance metrics for histogram data are shown and discussed. Note to Practitioners —This paper was motivated by the collaborative budget allocation (CBA) problem in which either a user has limited ratings that can be distributed among items, or a service has limited availability to all users in the sharing economy system. Existing approaches rarely consider this phenomenon in their collaborative prediction modeling, which implies the geometric constraints on a user’s rating vector. This paper suggests a matrix factorization framework to address this problem by learning a group of a user profile as basis points that can be combined to recover other users’ rating vectors. Considering the noise in observed ratings in real applications, we enhance the robustness of collaborative prediction by relaxing the recovery constraint. We then propose to specify the distance metric on the simplex from two perspectives, and give the optimization approach. In the future research, we will address the CBA in more complex settings, for example, some items can be substitutes or complementary for each other, in this way, how items can be better distributed among users is an interesting problem.