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）旨在通过两个非负低秩矩阵W和H的锥组合来逼近给定的非负矩阵M。传统上，NMF通过优化评估逼近质量的特定目标函数来求解，该评估通常基于Frobenius范数。本研究认为，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锥，NMF的这种锥视角可能与传统的点对点距离度量存在固有偏差。因此，本文提出采用射线对射线的弦距离作为度量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。我们通过将问题映射到流形上，应用黎曼优化技术求解弦距离NMF。与现有要求流形光滑的黎曼优化研究不同，弦距离NMF中的非负性构成不可微流形。我们提出黎曼乘法更新（RMU）方法，在保持黎曼梯度下降收敛特性的同时不破坏流形的光滑性条件。我们在合成数据集和真实多光谱图像上验证了弦距离NMF的有效性。