Nonnegative Matrix Factorization is an important tool in unsupervised machine learning to decompose a data matrix into a product of parts that are often interpretable. Many algorithms have been proposed during the last three decades. A well-known method is the Multiplicative Updates algorithm proposed by Lee and Seung in 2002. Multiplicative updates have many interesting features: they are simple to implement and can be adapted to popular variants such as sparse Nonnegative Matrix Factorization, and, according to recent benchmarks, is state-of-the-art for many problems where the loss function is not the Frobenius norm. In this manuscript, we propose to improve the Multiplicative Updates algorithm seen as an alternating majorization minimization algorithm by crafting a tighter upper bound of the Hessian matrix for each alternate subproblem. Convergence is still ensured and we observe in practice on both synthetic and real world dataset that the proposed fastMU algorithm is often several orders of magnitude faster than the regular Multiplicative Updates algorithm, and can even be competitive with state-of-the-art methods for the Frobenius loss.

翻译：非负矩阵分解是无监督机器学习中的重要工具，可将数据矩阵分解为通常可解释的局部乘积。在过去三十年间，已提出大量算法。其中一种著名方法是Lee和Seung于2002年提出的乘法更新算法。乘法更新具有诸多有趣特性：实现简单，可适配于稀疏非负矩阵分解等流行变体，且根据近期基准测试，在损失函数非Frobenius范数的诸多问题中仍属前沿方法。本文提出对交替性Majorization-Minimization优化框架下的乘法更新算法进行改进，通过为每个交替子问题构造更紧的Hessian矩阵上界来实现。该方法在保证收敛性的前提下，在合成数据集与真实数据集上的实验表明，所提出的fastMU算法通常比常规乘法更新算法快数个数量级，甚至在Frobenius损失函数下可与前沿方法竞争。