Matrix Factorization for Collaborative Budget Allocation

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.