Ordinal embedding (OE) aims to project objects into a low-dimensional space while preserving their ordinal constraints as well as possible. Generally speaking, a reasonable OE algorithm should simultaneously capture a) semantic meaning and b) the ordinal relationship of the objects. However, most of the existing methods merely focus on b). To address this issue, our goal in this paper is to seek a generic OE method to embrace the two features simultaneously. We argue that different dimensions of vector-based embedding are naturally entangled with each other. To realize a), we expect to decompose the D dimensional embedding space into D different semantic subspaces, where each subspace is associated with a matrix representation. Unfortunately, introducing a matrix-based representation requires far more complex parametric space than its vector-based counterparts. Thanks to the algebraic property of quaternions, we are able to find a more efficient way to represent a matrix with quaternions. For b), inspired by the classic chordal Grassmannian distance, a new distance function is defined to measure the distance between different quaternions/matrices, on top of which we construct a generic OE loss function. Experimental results for different tasks on both simulated and real-world datasets verify the effectiveness of our proposed method.

Heterogeneous information networks (HINs) have become a popular tool to capture complicated user-item relationships in recommendation problems in recent years. As a typical instantiation of HINs, meta-path is introduced in search of higher-level representations of user-item interactions. Though remarkable success has been achieved along this direction, existing meta-path-based recommendation methods face at least one of the following issues: 1) existing methods merely adopt simple meta-path fusion rules, which might be insufficient to exclude inconsistent information of different meta-paths that may hurt model performance; 2) the representative power is limited by shallow/stage-wise formulations. To solve these issues, we propose an end-to-end and unified embedding-based recommendation framework with graph-based learning. To address 1), we propose a flexible fusion module to integrate meta-path-based similarities into relative similarities between users and items. To address 2), we take advantage of the powerful representative ability of deep neural networks to learn more complicated and flexible latent embeddings. Finally, empirical studies on real-world datasets demonstrate the effectiveness of our proposed method.

The well-known collaborative filtering (CF) models typically optimize a single objective summed over all historical user-item interactions. Due to inevitable imbalances and biases in real-world data, they may develop a policy that unfairly discriminates against certain subgroups with low sample frequencies. To balance overall recommendation performance and fairness, prevalent solutions apply fairness constraints or regularizations to enforce equality of certain performance across different subgroups. However, simply enforcing equality of performance may lead to large performance degradation of those advantaged subgroups. To address this issue, we formulate a constrained Multi-Objective Optimization (MOO) problem. In contrast to the single objective, we treat the performance of each subgroup equivalently as an objective. This ensures that the imbalanced subgroup sample frequency does not affect the gradient information. We further propose fairness constraints to limit the search space to obtain more balanced solutions. To solve the constrained MOO problem, a gradient-based constrained MOO algorithm is proposed to seek a proper Pareto optimal solution for the performance trade-off. Extensive experiments on synthetic and real-world datasets show that our approach could help improve the recommendation accuracy of disadvantaged groups, while not damaging the overall performance.