Grassmannian Learning: Embedding Geometry Awareness in Shallow and Deep Learning
It addresses the problem of leveraging geometric structures in machine learning for practitioners in fields like computer vision and NLP, though it is incremental as it surveys existing approaches rather than proposing new methods.
The paper introduces Grassmannian learning as an emerging area that embeds geometry awareness from the Grassmann manifold into shallow and deep learning algorithms to tackle problems involving subspace-structured features, orthogonality constraints, or low-rank objectives, noting substantial performance improvements in various applications.
Modern machine learning algorithms have been adopted in a range of signal-processing applications spanning computer vision, natural language processing, and artificial intelligence. Many relevant problems involve subspace-structured features, orthogonality constrained or low-rank constrained objective functions, or subspace distances. These mathematical characteristics are expressed naturally using the Grassmann manifold. Unfortunately, this fact is not yet explored in many traditional learning algorithms. In the last few years, there have been growing interests in studying Grassmann manifold to tackle new learning problems. Such attempts have been reassured by substantial performance improvements in both classic learning and learning using deep neural networks. We term the former as shallow and the latter deep Grassmannian learning. The aim of this paper is to introduce the emerging area of Grassmannian learning by surveying common mathematical problems and primary solution approaches, and overviewing various applications. We hope to inspire practitioners in different fields to adopt the powerful tool of Grassmannian learning in their research.