Nonnegative Matrix Factorization (NMF) is the problem of approximating a given nonnegative matrix M through the conic combination of two nonnegative low-rank matrices W and H. Traditionally NMF is tackled by optimizing a specific objective function evaluating the quality of the approximation. This assessment is often done based on the Frobenius norm. In this study, we argue that the Frobenius norm as the "point-to-point" distance may not always be appropriate. Due to the nonnegative combination resulting in a polyhedral cone, this conic perspective of NMF may not naturally align with conventional point-to-point distance measures. Hence, a ray-to-ray chordal distance is proposed as an alternative way of measuring the discrepancy between M and WH. This measure is related to the Euclidean distance on the unit sphere, motivating us to employ nonsmooth manifold optimization approaches. We apply Riemannian optimization technique to solve chordal-NMF by casting it on a manifold. Unlike existing works on Riemannian optimization that require the manifold to be smooth, the nonnegativity in chordal-NMF is a non-differentiable manifold. We propose a Riemannian Multiplicative Update (RMU) that preserves the convergence properties of Riemannian gradient descent without breaking the smoothness condition on the manifold. We showcase the effectiveness of the Chordal-NMF on synthetic datasets as well as real-world multispectral images.

翻译：非负矩阵分解（NMF）是将给定非负矩阵M通过两个非负低秩矩阵W和H的锥组合进行近似的问题。传统上，NMF通过优化评估近似质量的特定目标函数来解决。这种评估通常基于Frobenius范数。本研究中，我们论证了作为"点对点"距离的Frobenius范数并非总是适用。由于非负组合形成多面体锥，这种NMF的锥视角可能无法自然适应传统的点对点距离度量。因此，我们提出一种射线到射线的弦距离，作为测量M与WH之间差异的替代方法。该度量与单位球面上的欧氏距离相关，促使我们采用非光滑流形优化方法。我们应用黎曼优化技术，将弦距离NMF问题转化为流形上的优化问题。与现有要求流形光滑的黎曼优化工作不同，弦距离NMF中的非负性构成不可微流形。我们提出了一种黎曼乘性更新（RMU）方法，该方法能在不破坏流形光滑性条件的前提下，保持黎曼梯度下降的收敛特性。我们在合成数据集及实际多光谱图像上展示了弦距离NMF的有效性。