Matrix factorization has been widely applied to various applications. With the fast development of storage and internet technologies, we have been witnessing a rapid increase of data. In this paper, we propose new algorithms for matrix factorization with the emphasis on efficiency. In addition, most existing methods of matrix factorization only consider a general smooth least square loss. Differently, many real-world applications have distinctive characteristics. As a result, different losses should be used accordingly. Therefore, it is beneficial to design new matrix factorization algorithms that are able to deal with both smooth and non-smooth losses. To this end, one needs to analyze the characteristics of target data and use the most appropriate loss based on the analysis. We particularly study two representative cases of low-rank matrix recovery, i.e., collaborative filtering for recommendation and high dynamic range imaging. To solve these two problems, we respectively propose a stage-wise matrix factorization algorithm by exploiting manifold optimization techniques. From our theoretical analysis, they are both are provably guaranteed to converge to a stationary point. Extensive experiments on recommender systems and high dynamic range imaging demonstrate the satisfactory performance and efficiency of our proposed method on large-scale real data.

中文翻译：

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快速和低内存成本矩阵分解：算法、分析和案例研究

矩阵分解已广泛应用于各种应用。随着存储和互联网技术的快速发展，我们见证了数据的快速增长。在本文中，我们提出了新的矩阵分解算法，重点是效率。此外，大多数现有的矩阵分解方法只考虑一般的平滑最小二乘损失。不同的是，许多现实世界的应用程序具有鲜明的特征。因此，应相应地使用不同的损耗。因此，设计能够处理平滑和非平滑损失的新矩阵分解算法是有益的。为此，需要分析目标数据的特征，并在分析的基础上使用最合适的损失。我们特别研究了低秩矩阵恢复的两个代表性案例，即推荐的协同过滤和高动态范围成像。为了解决这两个问题，我们分别提出了一种利用流形优化技术的阶段式矩阵分解算法。从我们的理论分析来看，它们都可以证明收敛到一个静止点。在推荐系统和高动态范围成像上的大量实验证明了我们提出的方法在大规模真实数据上的令人满意的性能和效率。可以证明它们都可以保证收敛到一个静止点。在推荐系统和高动态范围成像上的大量实验证明了我们提出的方法在大规模真实数据上的令人满意的性能和效率。可以证明它们都可以保证收敛到一个静止点。在推荐系统和高动态范围成像上的大量实验证明了我们提出的方法在大规模真实数据上的令人满意的性能和效率。