(Reproducing) kernel tricks

WARNING: This is very old. If I were to write it now, I would write it differently, and specifically more pedagogically.

Kernel in the sense of the “kernel trick”. Not to be confused with smoothing-type convolution kernels, nor the dozens of related-but-slightly-different clashing definitions of kernel; those can have their own respective pages. Corollary: If you do not know what to name something, call it a kernel.

We are concerned with a particular flavour of kernel in Hilbert spaces, specifically reproducing or Mercer kernels (Mercer 1909). The associated function space is a reproducing Kernel Hilbert Space, which is hereafter an RKHS.

Kernel tricks comprise the application of Mercer kernels in Machine Learning. The “trick” part is that many machine learning algorithms operate on inner products. Or can be rewritten to work that way. Such algorithms permit one to swap out a boring classic Euclidean definition of that inner product in favour of a fancy RKHS one. The classic machine learning pitch for trying such a stunt is something like “upgrade your old boring linear algebra on finite (usually low-) dimensional spaces to sexy algebra on potentially-infinite-dimensional feature spaces, which still has a low-dimensional representation.” Or, if you’d like, “apply certain statistical learning methods based on things with an obvious finite vector space representation (\(\mathbb{R}^n\)) to things without one (Sentences, piano-rolls, \(\mathcal{C}^d_\ell\)).”

Mini history: The oft-cited origins of all the reproducing kernel stuff are (Aronszajn 1950; Mercer 1909). It took a while to percolate into random function theory (Khintchine 1934; Yaglom 1987b) as covariance functions. Thence the idea arrived in statistical inference (Emanuel. Parzen 1962; E. Parzen 1963, 1959) and signal processing (Aasnaes and Kailath 1973; Duttweiler and Kailath 1973a, 1973b; Gevers and Kailath 1973; T. Kailath and Geesey 1971, 1973; T. Kailath 1971b, 1971a, 1974; T. Kailath, Geesey, and Weinert 1972; T. Kailath and Duttweiler 1972; T. Kailath and Weinert 1975), and now it is ubiquitous.

Practically, kernel methods have problems with scalability to large data sets. To apply any such method you need to keep a full Gram matrix of inner products between every data point, which needs you to know, for \(N\) data points, \(N(N-1)/2\) entries of a symmetric matrix. If you need to invert that matrix the cost is \(\mathcal{O}(N^3)\), which means you need fancy tricks to handle large \(N\). Fancy tricks depend on what the actual model is, but include Sparse GPs, random-projection inversions, Markov approximations and presumably many more

I’m especially interested in the application of such tricks in

1. kernel regression
2. wide random NNs
3. Nonparametric kernel independence tests
4. Efficient kernel pre-image approximation
5. Connection between kernel PCA and clustering (Schölkopf et al. 1998; Williams 2001) Turns out not all those applications are interesting to me.

1 Introductions

There are many primers on Mercer kernels and their connection to ML. Kenneth Tay’s intro is punchy. Il Shan Ng, Reproducing Kernel Hilbert Spaces & Machine Learning is good. See (Schölkopf and Smola 2002), which grinds out many connections with learning theory, or (Manton and Amblard 2015), which is more narrowly focused on just the Mercer-kernel part, and the topological and geometric properties of the spaces. (Ghojogh et al. 2021; Gori and Martínez-Herrero 2021; Gretton 2019). Cheney and Light (2009) is an approximation-theory perspective which does not especially concern itself with stochastic processes. I also seem to have bookmarked the following introductions (Vert, Tsuda, and Schölkopf 2004; Schölkopf et al. 1999; Schölkopf, Herbrich, and Smola 2001; Muller et al. 2001; Schölkopf and Smola 2003).

Alex Smola (who with, Bernhard Schölkopf) has his name on an intimidating proportion of publications in this area, also has all his publications online.