A Novel Framework for Learning Geometry-Aware Kernels

The data from real world usually have nonlinear geometric structure, which are often assumed to lie on or close to a low-dimensional manifold in a high-dimensional space. How to detect this nonlinear geometric structure of the data is important for the learning algorithms. Recently, there has been a surge of interest in utilizing kernels
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to exploit the manifold structure of the data. Such kernels are called geometry-aware kernels and are widely used in the machine learning algorithms. The performance of these algorithms critically relies on the choice of the geometry-aware kernels. Intuitively, a good geometryaware kernel should utilize additional information other than the geometric information. In many applications, it is required to compute the out-of-sample data directly. However, most of the geometry-aware kernel methods are restricted to the available data given beforehand, with no straightforward extension for out-of-sample data. In this paper, we propose a framework for more general geometry-aware kernel learning. The proposed framework integrates multiple sources of information and enables us to develop flexible and effective kernel matrices. Then, we theoretically show how the learned kernel matrices are extended to the corresponding kernel functions, in which the out-of-sample data can be computed directly. Under our framework, a novel family of geometry-aware kernels is developed. Especially, some existing geometry-aware kernels can be viewed as instances of our framework. The performance of the kernels is evaluated on dimensionality reduction, classification, and clustering tasks. The empirical results show that our kernels significantly improve the performance.