Nonnegative matrix factorization (NMF) is a linear dimensionality reduction technique for analyzing nonnegative data. A key aspect of NMF is the choice of the objective function that depends on the noise model (or statistics of the noise) assumed on the data. In many applications, the noise model is unknown and difficult to estimate. In this paper, we define a multi-objective NMF (MO-NMF) problem, where several objectives are combined within the same NMF model. We propose to use Lagrange duality to judiciously optimize for a set of weights to be used within the framework of the weighted-sum approach, that is, we minimize a single objective function which is a weighted sum of the all objective functions. We design a simple algorithm using multiplicative updates to minimize this weighted sum. We show how this can be used to find distributionally robust NMF solutions, that is, solutions that minimize the largest error among all objectives. We illustrate the effectiveness of this approach on synthetic, document and audio datasets. The results show that DR-NMF is robust to our incognizance of the noise model of the NMF problem.
翻译:非负矩阵因子化(NMF)是用于分析非负数据的一种线性维度减少技术。NMF的一个关键方面是选择取决于数据所假定的噪音模型(或噪音统计)的客观功能。在许多应用中,噪音模型是未知的,也难以估计。在本文件中,我们定义了一个多目标NMF(MO-NMF)问题,其中将几个目标合并在同一NMF模型中。我们提议使用拉格双轨法,明智地优化在加权和加权方法框架内使用的一套加权权重,也就是说,我们尽量减少一个单一目标函数,这是所有目标函数的加权总和。我们设计了一个简单的算法,使用倍增法来尽量减少这一加权总和。我们表明,如何利用这一算法找到分布上强大的NMF(MO-NMF)解决方案,即最大限度地减少所有目标中的最大错误。我们举例说明了这一方法在合成、文件和音频数据集方面的有效性。结果显示,DR-NMF对于我们了解NMF问题的噪音模型非常有力。