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Kernel methods offer the flexibility to learn complex relationships in modern, large data sets while enjoying strong theoretical guarantees on quality. Unfortunately, these methods typically require cubic running time in the data set size, a prohibitive cost in the large-data setting. Random feature maps (RFMs) and the Nyström method both consider low-rank approximations to the kernel matrix as a potential solution. But, in order to achieve desirable theoretical guarantees, the former may require a prohibitively large number of features J+, and the latter may be prohibitively expensive for high-dimensional problems. We propose to combine the simplicity and generality of RFMs with a data-dependent feature selection scheme to achieve desirable theoretical approximation properties of Nyström with just O(\log J+) features. Our key insight is to begin with a large set of random features, then reduce them to a small number of weighted features in a data-dependent, computationally efficient way, while preserving the statistical guarantees of using the original large set of features. We demonstrate the efficacy of our method with theory and experiments-including on a data set with over 50 million observations. In particular, we show that our method achieves small kernel matrix approximation error and better test set accuracy with provably fewer random features than state-of-the-art methods.
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Data‐driven linear complexity low‐rank approximation of general kernel matrices: A geometric approach
Summary A general, rectangular kernel matrix may be defined as where is a kernel function and where and are two sets of points. In this paper, we seek a low‐rank approximation to a kernel matrix where the sets of points and 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 corresponds to the training data and 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.
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- PAR ID:
- 10465994
- Date Published:
- Journal Name:
- Numerical Linear Algebra with Applications
- ISSN:
- 1070-5325
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1002/nla.2519
