Joint Normalization and Dimensionality Reduction on Grassmannian: A Generalized Perspective

This letter proposes a generalized framework with joint normalization that learns lower dimensional subspaces with maximum discriminative power by using Riemannian geometry. We model the similarity/dissimilarity between subspaces using various metrics defined on Grassmannian and formulate dimensionality reduction as a nonlinear constraint optimization problem considering the orthogonalization. To obtain the linear mapping, we derive the components required to perform Riemannian optimization 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 approach, any metric on the Grassmann manifold can theoretically reside in our model. We combine five metrics with our model, and the learning process is 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.