Nonnegative Matrix Factorization (NMF) is the problem of approximating a given nonnegative matrix \({\varvec{M}}\) through the product of two nonnegative low-rank matrices \({\varvec{W}}\) and \({\varvec{H}}\) . 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 (F-norm). In this work, we argue that the F-norm, as the “point-to-point” distance, may not always be appropriate. Viewing from the perspective of cone, NMF may not naturally align with F-norm. So, a ray-to-ray chordal distance is proposed as an alternative way of measuring the quality of the approximation. As this measure corresponds to the Euclidean distance on the sphere, it motivates the use of manifold optimization techniques. We apply Riemannian optimization technique to solve chordal-NMF by casting it on a manifold. Unlike works on Riemannian optimization that require the manifold to be smooth, the nonnegativity in chordal-NMF defines a non-differentiable manifold. We propose a Riemannian Multiplicative Update (RMU), and showcase the effectiveness of the chordal-NMF on synthetic and real-world datasets.
