Subspace-based learning with grassmann kernels
Daniel D. Lee, Jihun Hamm · 2008
In this thesis I propose a subspace-based learning paradigm for solving novel problems in machine learning. We often encounter subspace structures within data that lie inside a vector space. For example, the set of images of an object or a face under varying lighting conditions are known to lie on a low (4 or 9)-dimensional subspace with mild assumptions. Many other types of variations such as pose change or facial expression, can also be approximated quite well with low-dimensional subspaces. Treating such subspaces as basic units of learning gives rise to challenges that conventional algorithms cannot handle well. In this work, I tackle subspace-based learning problems with the unifying framework of Grassmann manifold, which is the set of linear subspaces of a Euclidean space. I propose positive definite kernels on this space, which provide an easy access to the repository of various kernel algorithms. Furthermore, I show that the Grassmann kernels can be extended to the set of affine and scaled subspaces. This extension allows us to handle larger classes of problems with little additional cost. The proposed kernels in this thesis can be used with any kernel method. In particular, I demonstrate the potential advantages of the proposed kernel with Discriminant Analysis techniques and Support Vector Machines for recognition and categorization tasks. Experiments with real image databases show not only the feasibility of the proposed framework, but also the improved performance of the method compared with previously known methods.