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A general, {\em 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 thispaper, we seek a low-rank approximation to a kernel matrix where the sets ofpoints $X$ and $Y$ are large and are arbitrarily distributed, such as away fromeach other, ``intermingled'', identical, etc. Such rectangular kernel matricesmay arise, for example, in Gaussian process regression where $X$ corresponds tothe training data and $Y$ corresponds to the test data. In this case, thepoints are often high-dimensional. Since the point sets are large, we mustexploit the fact that the matrix arises from a kernel function, and avoidforming the matrix, and thus ruling out most algebraic techniques. Inparticular, we seek methods that can scale linearly or nearly linear withrespect to the size of data for a fixed approximation rank. The main idea inthis paper is to {\em geometrically} select appropriate subsets of points toconstruct a low rank approximation. An analysis in this paper guides how thisselection should be performed.

Data-Driven Linear Complexity Low-Rank Approximation of General Kernel Matrices: A Geometric Approach