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 dimen-sionality reduction as a non-linear constraint optimization problem considering the orthogonalization. To obtain the linear mapping, we derive the components required to per-form Riemannian optimization (e.g., Riemannian conju-gate gradient) from the original Grassmannian through an orthonormal projection. We respect the Riemannian ge-ometry of the Grassmann manifold and search for this projection directly from one Grassmann manifold to an-other face-to-face without any additional transformations. In this natural geometry-aware way, any metric on the Grassmann manifold can be resided in our model theoreti-cally. We have combined five metrics with our model and the learning process can be treated as an unconstrained optimization problem on a Grassmann manifold. Exper-iments on several datasets demonstrate that our approach leads to a significant accuracy gain over state-of-the-art methods.