Data-driven linear complexity low-rank approximation of general kernel matrices: A geometric approach

A general, rectangular kernel matrix may be defined as $K_{ij}=\kappa (x_i,y_j)$ where $\kappa (x,y)$ is a kernel function and where $X={\{x_i\}}_{i=1}^m$ and $Y={\{y_i\}}_{i=1}^n$ are two sets of points. In this paper, we seek a low-rank approximation to a kernel matrix where the sets of points $X$ and $Y$ are large and are arbitrarily distributed, such as away from each other, “intermingled”, identical, and so forth. Such rectangular kernel matrices may arise, for example, in Gaussian process regression where $X$ corresponds to the training data and $Y$ corresponds to the test data. In this case, the points are often high-dimensional. Since the point sets are large, we must exploit the fact that the matrix arises from a kernel function, and avoid forming the matrix, and thus ruling out most algebraic techniques. In particular, we seek methods that can scale linearly or nearly linearly with respect to the size of data for a fixed approximation rank. The main idea in this paper is to geometrically select appropriate subsets of points to construct a low rank approximation. An analysis in this paper guides how this selection should be performed.