Joint interaction with context operation for collaborative filtering

In recommender systems, the classical matrix factorization model for collaborative filtering only considers joint interactions between users and items. In contrast, context-aware recommender systems (CARS) use contexts to improve recommendation performance. Some early CARS models treat user, item and context equally, unable to capture contextual impact accurately. More recent models perform context operations on users and items separately, leading to “double-counting” of contextual information. This paper proposes a new model, Joint Interaction with Context Operation (JICO), to integrate the joint interaction model with the context operation model, via two layers. The joint interaction layer models interactions between users and items via an interaction tensor. The context operation layer captures contextual information via a contextual operating tensor. We evaluate JICO on four datasets and conduct novel studies, including varying contextual influence and time split recommendation. JICO consistently outperforms competing methods, while providing many useful insights to assist further analysis

The cost of continuity: performance of iterative solvers on isogeometric finite elements

In this paper we study how the use of a more continuous set of basis functions affects the cost of solving systems of linear equations resulting from a discretized Galerkin weak form. Specifically, we compare performance of linear solvers when discretizing using $C^0$ B-splines, which span traditional finite element spaces, and $C^{p-1}$ B-splines, which represent maximum continuity. We provide theoretical estimates for the increase in cost of the matrix-vector product as well as for the construction and application of black-box preconditioners. We accompany these estimates with numerical results and study their sensitivity to various grid parameters such as element size $h$ and polynomial order of approximation $p$. Finally, we present timing results for a range of preconditioning options for the Laplace problem. We conclude that the matrix-vector product operation is at most \slfrac{33p^2}{8} times more expensive for the more continuous space, although for moderately low $p$, this number is significantly reduced. Moreover, if static condensation is not employed, this number further reduces to at most a value of 8, even for high $p$. Preconditioning options can be up to $p^3$ times more expensive to setup, although this difference significantly decreases for some popular preconditioners such as Incomplete LU factorization