Dimensionality Reduction on Grassmannian via Riemannian Optimization: A Generalized Perspective

This paper proposes a generalized framework with joint normalization which learns lower-dimensional subspaces with maximum discriminative power by making use of the Riemannian geometry. In particular, we model the similarity/dissimilarity between subspaces using various metrics defined on Grassmannian and formulate dimensionality reduction as a non-linear constraint optimization problem considering the orthogonalization. To obtain the linear mapping, we derive the components required to perform Riemannian optimization (e.g., Riemannian conjugate gradient) from the original Grassmannian through an orthonormal projection. We respect the Riemannian geometry of the Grassmann manifold and search for this projection directly from one Grassmann manifold to another face-to-face without any additional transformations. In this natural geometry-aware way, any metric on the Grassmannmanifold can be resided in our model theoretically. We have combined five metrics with our model and the learning process can be treated as an unconstrained optimization problem on a Grassmann manifold. Experiments on several datasets demonstrate that our approach leads to a significant accuracy gain over state-of-the-art methods.