Nonnegative matrix factorization (NMF) is one of the best-known multivariate data analysis techniques. The NMF uniqueness and its rank selection are two major open problems in this field. The solutions uniqueness issue can be addressed by imposing the orthogonality condition on NMF. This constraint yields sparser part-based representations and improved performance in clustering and source separation tasks. However, existing orthogonal NMF algorithms rely mainly on non-probabilistic frameworks that ignore the noise inherent in real-life data and lack variable uncertainties. Thus, in this work, we investigate a new probabilistic formulation of orthogonal NMF (ONMF). In the proposed model, we impose the orthogonality through a directional prior distribution defined on the Stiefel manifold called von Mises-Fisher distribution. This manifold consists of a set of directions that comply with the orthogonality condition that arises in many applications. Moreover, our model involves an automatic relevance determination (ARD) prior to address the model order selection issue. We devised an efficient variational Bayesian inference algorithm to solve the proposed ONMF model, which allows fast processing of large datasets. We evaluated the proposed model, called VBONMF, on the task of blind decomposition of real-world multispectral images of ancient documents. The numerical experiments demonstrate its efficiency and competitiveness compared to the state-of-the-art approaches.

中文翻译：

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Stiefel 流形上的变分贝叶斯正交非负矩阵分解


非负矩阵分解 (NMF) 是最著名的多元数据分析技术之一。 NMF 的独特性及其等级选择是该领域的两个主要开放问题。解决方案的唯一性问题可以通过对 NMF 施加正交条件来解决。此约束产生更稀疏的基于部分的表示，并提高了聚类和源分离任务的性能。然而，现有的正交 NMF 算法主要依赖于非概率框架，该框架忽略了现实数据中固有的噪声并且缺乏可变的不确定性。因此，在这项工作中，我们研究了正交 NMF (ONMF) 的新概率公式。在所提出的模型中，我们通过在 Stiefel 流形上定义的方向先验分布（称为 von Mises-Fisher 分布）来施加正交性。该流形由一组符合许多应用中出现的正交条件的方向组成。此外，我们的模型在解决模型顺序选择问题之前涉及自动相关性确定（ARD）。我们设计了一种有效的变分贝叶斯推理算法来解决所提出的 ONMF 模型，该模型允许快速处理大型数据集。我们评估了所提出的名为 VBONMF 的模型，用于对古代文献的真实多光谱图像进行盲分解的任务。数值实验证明了其与最先进的方法相比的效率和竞争力。