Non‐negative matrix factorisation (NMF) is an increasingly popular unsupervised learning method. However, parameter estimation in the NMF model is a difficult high‐dimensional optimisation problem. We consider algorithms of the alternating least squares type. Solutions to the least squares problem fall in two categories. The first category is iterative algorithms, which include algorithms such as the majorise–minimise (MM) algorithm, coordinate descent, gradient descent and the Févotte‐Cemgil expectation–maximisation (FC‐EM) algorithm. We introduce a new family of iterative updates based on a generalisation of the FC‐EM algorithm. The coordinate descent, gradient descent and FC‐EM algorithms are special cases of this new EM family of iterative procedures. Curiously, we show that the MM algorithm is never a member of our general EM algorithm. The second category is based on cone projection. We describe and prove a cone projection algorithm tailored to the non‐negative least square problem. We compare the algorithms on a test case and on the problem of identifying mutational signatures in human cancer. We generally find that cone projection is an attractive choice. Furthermore, in the cancer application, we find that a mix‐and‐match strategy performs better than running each algorithm in isolation.

中文翻译：

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非负矩阵分解的统一框架和算法比较

非负矩阵分解（NMF）是一种越来越流行的无监督学习方法。但是，NMF模型中的参数估计是一个困难的高维优化问题。我们考虑交替最小二乘类型的算法。最小二乘问题的解决方案分为两类。第一类是迭代算法，其中包括诸如最小化（MM）算法，坐标下降，梯度下降和Févotte-Cemgil期望最大化（FC-EM）算法之类的算法。我们基于FC‐EM算法的泛化引入了一个新的迭代更新系列。坐标下降，梯度下降和FC‐EM算法是这种新的EM迭代过程系列的特殊情况。奇怪的是，我们表明MM算法永远不会我们通用EM算法的成员。第二类基于圆锥投影。我们描述并证明了一种针对非负最小二乘问题的圆锥投影算法。我们在一个测试案例和识别人类癌症中的突变特征的问题上比较了算法。通常，我们发现圆锥投影是一个有吸引力的选择。此外，在癌症应用中，我们发现混合匹配策略比单独运行每个算法要好。