非负矩阵分解的高性能并行算法
A High-Performance Parallel Algorithm for Nonnegative Matrix Factorization
摘要：
非负矩阵分解（NMF）是针对给定的输入矩阵$A$确定两个非负低秩因子$W$和$H$的问题，使得$A \approx W H$。NMF在文本挖掘中的主题建模、视频分析中的背景分离、社交网络中的社区发现等领域有着广泛的应用。尽管它在数据采掘社区中很受欢迎，但缺乏有效的并行软件来解决大数据集的问题。现有的分布式内存算法在性能和适用性方面受到限制，因为它们是使用Hadoop实现的，并且仅针对稀疏矩阵设计。我们提出了一个分布式存储器并行算法，通过迭代求解交替的非负最小二乘（NLS）子问题的$W$和$H$的因式分解计算。据我们所知，我们的算法是第一个高性能的并行算法NMF。它在内存中维护数据和因子矩阵（分布在处理器中），使用MPI进行处理器间通信，并且在密集的情况下，可以证明最小化通信成本（在温和的假设下）。与以前的实现相反，我们的算法也很灵活：（1）它对密集和稀疏矩阵都有很好的表现，（2）它允许用户从多个算法中选择，用于在交替迭代中解决局部NLS子问题。我们展示了我们的算法的可扩展性，并将其与基线实现进行比较，显示出显着的性能改进。
英文摘要：
Non-negative matrix factorization (NMF) is the problem of determining two non-negative low rank factors $W$ and $H$, for the given input matrix $A$, such that $A \approx W H$. NMF is a useful tool for many applications in different domains such as topic modeling in text mining, background separation in video analysis, and community detection in social networks. Despite its popularity in the data mining community, there is a lack of efficient parallel software to solve the problem for big datasets. Existing distributed-memory algorithms are limited in terms of performance and applicability, as they are implemented using Hadoop and are designed only for sparse matrices. We propose a distributed-memory parallel algorithm that computes the factorization by iteratively solving alternating non-negative least squares (NLS) subproblems for $W$ and $H$. To our knowledge, our algorithm is the first high-performance parallel algorithm for NMF. It maintains the data and factor matrices in memory (distributed across processors), uses MPI for interprocessor communication, and, in the dense case, provably minimizes communication costs (under mild assumptions). As opposed to previous implementations, our algorithm is also flexible: (1) it performs well for dense and sparse matrices, and (2) it allows the user to choose from among multiple algorithms for solving local NLS subproblems within the alternating iterations. We demonstrate the scalability of our algorithm and compare it with baseline implementations, showing significant performance improvements.
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