We consider the problem of approximating a general Gaussian location mixture by finite mixtures. The minimum order of finite mixtures that achieve a prescribed accuracy is determined within constant factors for the family of mixing distributions with compact support or appropriate assumptions on the tail probability including subgaussian and subexponential. While the upper bound is achieved using the technique of local moment matching, the lower bound is established by relating the best approximation error to the low-rank approximation of certain trigonometric moment matrices, followed by a refined spectral analysis of their minimum eigenvalue. In the case of Gaussian mixing distributions, this result corrects a previous lower bound in [2].

We consider the problem of approximating a general Gaussian location mixture by finite mixtures. The minimum order of finite mixtures that achieve a prescribed accuracy (measured by various f-divergences) are determined within constant factors for the family of compactly supported or subgaussian mixing distributions. While the upper bound is achieved using the technique of local moment matching, the lower bound is established by relating the best approximation error to the low-rank approximation of certain trigonometric moment matrices and weighted moment matrices, followed by a refined spectral analysis of the minimum eigenvalue of these matrices. In the case of Gaussian mixing distributions, this result corrects a previous lower bound in [1].

Implicit feedback is widely explored by modern recommender systems. Since the feedback is often sparse and imbalanced, it poses great challenges to the learning of complex interactions among users and items. Metric learning has been proposed to capture user-item interactions from implicit feedback, but existing methods only represent users and items in a single metric space, ignoring the fact that users can have multiple preferences and items can have multiple properties, which leads to potential conflicts limiting their performance in recommendation. To capture the multiple facets of user preferences and item properties while resolving their potential conflicts, we propose the novel framework of Multi-fAcet Recommender networks with Spherical optimization (MARS). By designing a cross-facet similarity measurement, we project users and items into multiple metric spaces for fine-grained representation learning, and compare them only in the proper spaces. Furthermore, we devise a spherical optimization strategy to enhance the effectiveness and robustness of the multi-facet recommendation framework. Extensive experiments on six real-world benchmark datasets show drastic performance gains brought by MARS, which constantly achieves up to 40% improvements over the state-of-the-art baselines regarding both HR and nDCG metrics.