Kernel Sketching yields Kernel JL
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The main contribution of the paper is to show that Gaussian sketching of a kernel-Gram matrix K yields an operator whose counterpart in an RKHS H, is a random projection operator—in the spirit of Johnson-Lindenstrauss (JL) lemma. To be precise, given a random matrix Z with i.i.d. Gaussian entries, we show that a sketch ZK corresponds to a particular random operator in (infinite-dimensional) Hilbert space H that maps functions f ∈H to a low-dimensional space R^d, while preserving a weighted RKHS inner-product of the form 〈 f, g 〉_Σ〈 f, Σ^3 g 〉_H, where Σ is the covariance operator induced by the data distribution. In particular, under similar assumptions as in kernel PCA (KPCA), or kernel k-means (K-k-means), well-separated subsets of feature-space {K(·, x): x ∈ X} remain well-separated after such operation, which suggests similar benefits as in KPCA and/or K-k-means, albeit at the much cheaper cost of a random projection. In particular, our convergence rates suggest that, given a large dataset {X_i}_i=1^N of size N, we can build the Gram matrix K on a much smaller subsample of size n≪ N, so that the sketch ZK is very cheap to obtain and subsequently apply as a projection operator on the original data {X_i}_i=1^N. We verify these insights empirically on synthetic data, and on real-world clustering applications.
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