A page where I document whet I *don’t* know about kernel approximation.
A page about what I *do* know would be empty.

What I mean is: approximating implicit Mercer kernel features with explicit features, that is; Equivalently, approximating the Gram matrix, which is also related to mixture model inference and clustering. I’m especially interested in how it might be done using random linear algebra. I am mostly interested in translation-invariant kernels, so assume I’m talking about those unless I say otherwise.

*Not* the related but distinct (terminological collision)
of approximating functions from mixtures of kernels.
Also note that the Fast Gauss Transform and other related fast multipole methods,
while commonly used to approximate convolution kernels,
can also approximate certain Mercer kernels but it’s not what I mean here -
fast multipole methods gives you an approximation to the field generated
by all kernels and all the support points
In kernel approximation we look for approximations, in some sense, to
*the component kernels themselves*.

Short introduction at scikit-learn kernel approximation page.

DrMa05, YLMJ12, VeZi12, LiIS10, RaRe07, AlMa14, Bach15, Bach13, YLMJ12, VVZJ10 have work here.

The approximations might be random projection, or random sampling based, e.g. the Nyström method, which is reportedly effectively Monte Carlo integration, although I understand there is an optimisation step too?

I need to work out the difference between random Fourier features, random kitchen sinks, Nyström methods and whatever Smola et al (SKSB98) call their special case Gaussian approximation. I think Random Fourier features are the same as random kitchen sinks, (RaRe07) and Nyström is different (WiSe01). When we can exploit (data- or kernel-) structure to do better? (say, LeSS13, VVZJ10) Quasi Monte Carlo can improve on random Monte Carlo? (update: someone already had that idea: YSAM14) Or better matrix factorisations?

Also I guess we need to know trade-offs of computational time/space cost versus approximation bounds, so that we can decide when to bother. When is it enough to reduce computational cost merely with support vectors, or to evaluate the kernels efficiently using, e.g. the Fast Gauss Transform and related methods methods, rather than coming up with alternative kernel bases? (e.g. you don’t want to do the fiddly coding for the Faust Gauss transform)

Does this help? Chris Ding’s Principal Component Analysis and Matrix Factorizations for Learning.

Recently, Maji and Berg and Vedaldi and Zisserman proposed explicit feature maps to approximate the additive kernels (intersection, \(\chi^2\), etc.) by linear ones, thus enabling the use of fast machine learning technique in a non-linear context. An analogous technique was proposed by Rahimi and Recht (RaRe07) for the translation invariant RBF kernels. In this paper, we complete the construction and combine the two techniques to obtain explicit feature maps for the generalized RBF kernels.

## Lectures: Fastfood etc

## Implementations

This is the LIBASKIT set of scalable machine learning and data analysis tools. Currently we provide codes for kernel sums, nearest-neighbors, kmeans clustering, kernel regression, and multiclass kernel logistic regression. All codes use OpenMP and MPI for shared memory and distributed memory parallelism.

[…] PNYSTR : (Parallel Nystrom method) Code for kernel summation using the Nystrom method.

## Connections

MDS and Kernel PCA and mixture models are all related in a way I should try to understand when I have a moment.

For the connection with kernel PCA, see Smola et al ( SKSB98) and Williams for metric multidimensional scaling ( Will01).

## Refs

- AlMa14
- Alaoui, A. E., & Mahoney, M. W.(2014) Fast Randomized Kernel Methods With Statistical Guarantees.
*arXiv:1411.0306 [Cs, Stat]*. - Bach15
- Bach, F. (2015) On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions.
*arXiv Preprint arXiv:1502.06800*. - Bach13
- Bach, F. R.(2013) Sharp analysis of low-rank kernel matrix approximations. In COLT (Vol. 30, pp. 185–209).
- BaWS04
- Bakır, G. H., Weston, J., & Schölkopf, B. (2004) Learning to find pre-images.
*Advances in Neural Information Processing Systems*, 16(7), 449–456. - BeMo10
- Beylkin, G., & Monzón, L. (2010) Approximation by exponential sums revisited.
*Applied and Computational Harmonic Analysis*, 28(2), 131–149. DOI. - BrBH16
- Brault, R., d’Alché-Buc, F., & Heinonen, M. (2016) Random Fourier Features for Operator-Valued Kernels.
*arXiv:1605.02536 [Cs]*. - BrLB00
- Brault, R., Lim, N., & d’Alché-Buc, F. (n.d.) Scaling up Vector Autoregressive Models With Operator-Valued Random Fourier Features.
- ChLi09
- Cheney, E. W., & Light, W. A.(2009) A Course in Approximation Theory. . American Mathematical Soc.
- ChSi16
- Choromanski, K., & Sindhwani, V. (2016) Recycling Randomness with Structure for Sublinear time Kernel Expansions.
*arXiv:1605.09049 [Cs, Stat]*. - CuSS08
- Cunningham, J. P., Shenoy, K. V., & Sahani, M. (2008) Fast Gaussian process methods for point process intensity estimation. (pp. 192–199). ACM Press DOI.
- CBMF16
- Cutajar, K., Bonilla, E. V., Michiardi, P., & Filippone, M. (2016) Practical Learning of Deep Gaussian Processes via Random Fourier Features.
*arXiv:1610.04386 [Stat]*. - DrMa05
- Drineas, P., & Mahoney, M. W.(2005) On the Nyström method for approximating a Gram matrix for improved kernel-based learning.
*Journal of Machine Learning Research*, 6, 2153–2175. - GlLi16
- Globerson, A., & Livni, R. (2016) Learning Infinite-Layer Networks: Beyond the Kernel Trick.
*arXiv:1606.05316 [Cs]*. - KwTs04
- Kwok, J. T.-Y., & Tsang, I. W.-H. (2004) The Pre-Image Problem in Kernel Methods.
*IEEE Transactions on Neural Networks*, 15(6), 1517–1525. DOI. - LeSS13
- Le, Q., Sarlós, T., & Smola, A. (2013) Fastfood-approximating kernel expansions in loglinear time. In Proceedings of the international conference on machine learning.
- LiIS10
- Li, F., Ionescu, C., & Sminchisescu, C. (2010) Random Fourier Approximations for Skewed Multiplicative Histogram Kernels. In M. Goesele, S. Roth, A. Kuijper, B. Schiele, & K. Schindler (Eds.), Pattern Recognition (pp. 262–271). Springer Berlin Heidelberg DOI.
- MiNY06
- Minh, H. Q., Niyogi, P., & Yao, Y. (2006) Mercer’s theorem, feature maps, and smoothing. In International Conference on Computational Learning Theory (pp. 154–168). Springer DOI.
- PoBe16a
- Pourkamali-Anaraki, F., & Becker, S. (2016a) A Randomized Approach to Efficient Kernel Clustering.
*arXiv:1608.07597 [Stat]*. - PoBe16b
- Pourkamali-Anaraki, F., & Becker, S. (2016b) Randomized Clustered Nystrom for Large-Scale Kernel Machines.
*arXiv:1612.06470 [Cs, Stat]*. - RaRe07
- Rahimi, A., & Recht, B. (2007) Random features for large-scale kernel machines. In Advances in neural information processing systems (pp. 1177–1184). Curran Associates, Inc.
- RaRe09
- Rahimi, A., & Recht, B. (2009) Weighted Sums of Random Kitchen Sinks: Replacing minimization with randomization in learning. In Advances in neural information processing systems (pp. 1313–1320). Curran Associates, Inc.
- SKSB98
- Schölkopf, B., Knirsch, P., Smola, A., & Burges, C. (1998) Fast Approximation of Support Vector Kernel Expansions, and an Interpretation of Clustering as Approximation in Feature Spaces. In P. Levi, M. Schanz, R.-J. Ahlers, & F. May (Eds.), Mustererkennung 1998 (pp. 125–132). Springer Berlin Heidelberg DOI.
- ScSm02
- Schölkopf, B., & Smola, A. J.(2002) Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. . MIT Press
- ScSM97
- Schölkopf, B., Smola, A., & Müller, K.-R. (1997) Kernel principal component analysis. In W. Gerstner, A. Germond, M. Hasler, & J.-D. Nicoud (Eds.), Artificial Neural Networks — ICANN’97 (pp. 583–588). Springer Berlin Heidelberg DOI.
- VeZi12
- Vedaldi, A., & Zisserman, A. (2012) Efficient Additive Kernels via Explicit Feature Maps.
*IEEE Transactions on Pattern Analysis and Machine Intelligence*, 34(3), 480–492. DOI. - VVZJ10
- Vempati, S., Vedaldi, A., Zisserman, A., & Jawahar, C. (2010) Generalized RBF feature maps for efficient detection. In BMVC (pp. 1–11).
- Will01
- Williams, C. K. I.(2001) On a Connection between Kernel PCA and Metric Multidimensional Scaling. In T. K. Leen, T. G. Dietterich, & V. Tresp (Eds.), Advances in Neural Information Processing Systems 13 (Vol. 46, pp. 675–681). MIT Press DOI.
- WiSe01
- Williams, C. K., & Seeger, M. (2001) Using the Nyström Method to Speed Up Kernel Machines. In Advances in Neural Information Processing Systems (pp. 682–688).
- YSAM14
- Yang, J., Sindhwani, V., Avron, H., & Mahoney, M. (2014) Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels.
*arXiv:1412.8293 [Cs, Math, Stat]*. - YLMJ12
- Yang, T., Li, Y.-F., Mahdavi, M., Jin, R., & Zhou, Z.-H. (2012) Nyström method vs random fourier features: A theoretical and empirical comparison. In Advances in neural information processing systems (pp. 476–484).
- YuMB17
- Yu, C. D., March, W. B., & Biros, G. (2017) An $N log N$ Parallel Fast Direct Solver for Kernel Matrices.
*arXiv:1701.02324 [Cs]*. - ZFGS16
- Zhang, Q., Filippi, S., Gretton, A., & Sejdinovic, D. (2016) Large-Scale Kernel Methods for Independence Testing.
*arXiv:1606.07892 [Stat]*.