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Techniques of matrix completion aim to impute a large portion of missing entries in a data matrix through a small portion of observed ones. In practice, prior information and special structures are usually employed in order to improve the accuracy of matrix completion. In this paper, we propose a unified nonconvex optimization framework for matrix completion with linearly parameterized factors. In particular, by introducing a condition referred to as Correlated Parametric Factorization, we conduct a unified geometric analysis for the nonconvex objective by establishing uniform upper bounds for low-rank estimation resulting from any local minimizer. Perhaps surprisingly, the condition of Correlated Parametric Factorization holds for important examples including subspace-constrained matrix completion and skew-symmetric matrix completion. The effectiveness of our unified nonconvex optimization method is also empirically illustrated by extensive numerical simulations. 

Recommender systems have been extensively used by the entertainment industry, business marketing and the biomedical industry. In addition to its capacity of providing preference-based recommendations as an unsupervised learning methodology, it has been also proven useful in sales forecasting, product introduction and other production related businesses. Since some consumers and companies need a recommendation or prediction for future budget, labor and supply chain coordination, dynamic recommender systems for precise forecasting have become extremely necessary. In this article, we propose a new recommendation method, namely the dynamic tensor recommender system (DTRS), which aims particularly at forecasting future recommendation. The proposed method utilizes a tensor-valued function of time to integrate time and contextual information, and creates a time-varying coefficient model for temporal tensor factorization through a polynomial spline approximation. Major advantages of the proposed method include competitive future recommendation predictions and effective prediction interval estimations. In theory, we establish the convergence rate of the proposed tensor factorization and asymptotic normality of the spline coefficient estimator. The proposed method is applied to simulations, IRI marketing data and Last.fm data. Numerical studies demonstrate that the proposed method outperforms existing methods in terms of future time forecasting. 

Neural networks provide a rich class of high-dimensional, non-convex optimization problems. Despite their non-convexity, gradient-descent methods often successfully optimize these models. This has motivated a recent spur in research attempting to characterize properties of their loss surface that may explain such success. In this paper, we address this phenomenon by studying a key topological property of the loss: the presence or absence of spurious valleys, defined as connected components of sub-level sets that do not include a global minimum. Focusing on a class of two-layer neural networks defined by smooth (but generally non-linear) activation functions, we identify a notion of intrinsic dimension and show that it provides necessary and sufficient conditions for the absence of spurious valleys. More concretely, finite intrinsic dimension guarantees that for sufficiently overparametrised models no spurious valleys exist, independently of the data distribution. Conversely, infinite intrinsic dimension implies that spurious valleys do exist for certain data distributions, independently of model overparametrisation. Besides these positive and negative results, we show that, although spurious valleys may exist in general, they are confined to low risk levels and avoided with high probability on overparametrised models.