Dimensionality Reduction on Grassmannian via Riemannian Optimization: A Generalized Perspective
This provides a generalized framework for dimensionality reduction that can incorporate various metrics, potentially benefiting machine learning applications in fields like computer vision or signal processing, though it appears incremental as it builds on existing Riemannian optimization techniques.
The paper tackles dimensionality reduction by learning discriminative subspaces using Riemannian geometry on Grassmann manifolds, achieving significant accuracy gains over state-of-the-art methods in experiments.
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.