Improved SVD-based Initialization for Nonnegative Matrix Factorization using Low-Rank Correction
Graduate School of Science and Engineering（研究生院）；PAF-Karachi Institute of Economics and Technology（卡拉奇经济和技术学院）；Department of Mathematics and Operational Research（数学与运筹学系）
Due to the iterative nature of most nonnegative matrix factorization (\textsc{NMF}) algorithms, initialization is a key aspect as it significantly influences both the convergence and the final solution obtained. Many initialization schemes have been proposed for NMF, among which one of the most popular class of methods are based on the singular value decomposition (SVD). However, these SVD-based initializations do not satisfy a rather natural condition, namely that the error should decrease as the rank of factorization increases. In this paper, we propose a novel SVD-based \textsc{NMF} initialization to specifically address this shortcoming by taking into account the SVD factors that were discarded to obtain a nonnegative initialization. This method, referred to as nonnegative SVD with low-rank correction (NNSVD-LRC), allows us to significantly reduce the initial error at a negligible additional computational cost using the low-rank structure of the discarded SVD factors. NNSVD-LRC has two other advantages compared to previous SVD-based initializations: (1) it provably generates sparse initial factors, and (2) it is faster as it only requires to compute a truncated SVD of rank $\lceil r/2 + 1 \rceil$ where $r$ is the factorization rank of the sought NMF decomposition (as opposed to a rank-$r$ truncated SVD for other methods). We show on several standard dense and sparse data sets that our new method competes favorably with state-of-the-art SVD-based initializations for NMF.
School of Computer Science and Engineering, Southeast University（东南大学计算机科学与工程学院）；City University of Hong Kong（香港城市大学）；City University of Hong Kong Shenzhen Research Institute（香港城市大学深圳研究院）
Adaptive Nonnegative Matrix Factorization and Measure Comparisons for Recommender Systems
Università di Pisa（比萨大学）
评价矩阵的非负矩阵分解（NMF）已被证明是解决推荐问题的有效方法。在本文中，我们提出了新的方法的基础上NMF的评级矩阵，我们比较了他们与一些经典的算法，如SVD和正则化和非正则化非负矩阵分解方法。特别地，给出了一种新的算法，该算法在每一步中自适应地改变待极小化的函数，实现了一种动态的先验策略。另一种算法是通过对先前项的未知评级进行重构来修改NMF公式中要最小化的函数。然后，我们结合不同的方法，得到两个混合策略，这是非常有效的重建丢失的意见。我们进行了周到的比较不同的方法的基础上的几个评价措施。我们认为，特别是评级，分类和排名的措施表明，算法获得最好的分数为一个给定的措施是一般最好的，也当考虑不同的措施，降低在设计具体的评估措施的兴趣。这些算法已经在不同的数据集上进行了测试，特别是包含电影评级的1M和10M MovieLens数据集，包含笑话评级的Jester数据集和包含食物评级的Amazon Fine Foods数据集。不同算法的比较，显示了良好的性能的方法，采用显式和隐式正则化计划。此外，我们可以得到一个快速的方法与更准确的一个相结合的混合策略的推动。
Subspace Learning for Feature Selection via Rank Revealing QR Factorization: Unsupervised and Hybrid Approaches with Non-negative Matrix Factorization and Evolutionary Algorithm
Toronto Metropolitan University（多伦多都会大学）；University of Toronto（多伦多大学）
