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Kernel approximation

Usefulness: 🔧
Novelty: 💡
Uncertainty: 🤪 🤪 🤪
Incompleteness: 🚧 🚧 🚧

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.

VVZJ10:

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

libaskit:

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

Alaoui, Ahmed El, and Michael W. Mahoney. 2014. “Fast Randomized Kernel Methods with Statistical Guarantees,” November. http://arxiv.org/abs/1411.0306.

Bach, Francis. 2015. “On the Equivalence Between Kernel Quadrature Rules and Random Feature Expansions.” arXiv Preprint arXiv:1502.06800. http://arxiv.org/abs/1502.06800.

Bach, Francis R. 2013. “Sharp Analysis of Low-Rank Kernel Matrix Approximations.” In COLT, 30:185–209. http://www.jmlr.org/proceedings/papers/v30/Bach13.pdf.

Bakır, Gökhan H., Jason Weston, and Bernhard Schölkopf. 2004. “Learning to Find Pre-Images.” Advances in Neural Information Processing Systems 16 (7): 449–56. http://www.kyb.mpg.de/fileadmin/user_upload/files/publications/pdf2262.pdf.

Beylkin, Gregory, and Lucas Monzón. 2010. “Approximation by Exponential Sums Revisited.” Applied and Computational Harmonic Analysis 28 (2): 131–49. https://doi.org/10.1016/j.acha.2009.08.011.

Brault, Romain, Florence d’Alché-Buc, and Markus Heinonen. 2016. “Random Fourier Features for Operator-Valued Kernels.” In Proceedings of the 8th Asian Conference on Machine Learning, 110–25. http://arxiv.org/abs/1605.02536.

Brault, Romain, Néhémy Lim, and Florence d’Alché-Buc. n.d. “Scaling up Vector Autoregressive Models with Operator-Valued Random Fourier Features.” Accessed August 31, 2016. https://aaltd16.irisa.fr/files/2016/08/AALTD16_paper_11.pdf.

Cheney, Elliott Ward, and William Allan Light. 2009. A Course in Approximation Theory. American Mathematical Soc. https://books.google.com.au/books?hl=en&lr=&id=II6DAwAAQBAJ&oi=fnd&pg=PA1&ots=ch9-LyxDg6&sig=jetWpIErExYvlnnSsup-5yHhso0.

Choromanski, Krzysztof, Mark Rowland, and Adrian Weller. 2017. “The Unreasonable Effectiveness of Random Orthogonal Embeddings,” March. http://arxiv.org/abs/1703.00864.

Choromanski, Krzysztof, and Vikas Sindhwani. 2016. “Recycling Randomness with Structure for Sublinear Time Kernel Expansions,” May. http://arxiv.org/abs/1605.09049.

Cunningham, John P., Krishna V. Shenoy, and Maneesh Sahani. 2008. “Fast Gaussian Process Methods for Point Process Intensity Estimation.” In Proceedings of the 25th International Conference on Machine Learning, 192–99. ICML ’08. New York, NY, USA: ACM Press. https://doi.org/10.1145/1390156.1390181.

Curto, Joachim, Irene Zarza, Feng Yang, Alexander J. Smola, and Luc Van Gool. 2017. “F2F: A Library for Fast Kernel Expansions,” February. http://arxiv.org/abs/1702.08159.

Cutajar, Kurt, Edwin V. Bonilla, Pietro Michiardi, and Maurizio Filippone. 2017. “Random Feature Expansions for Deep Gaussian Processes.” In PMLR. http://proceedings.mlr.press/v70/cutajar17a.html.

Drineas, Petros, and Michael W. Mahoney. 2005. “On the Nyström Method for Approximating a Gram Matrix for Improved Kernel-Based Learning.” Journal of Machine Learning Research 6 (December): 2153–75. http://jmlr.org/papers/volume6/drineas05a/drineas05a.pdf.

Globerson, Amir, and Roi Livni. 2016. “Learning Infinite-Layer Networks: Beyond the Kernel Trick,” June. http://arxiv.org/abs/1606.05316.

Kwok, J. T.-Y., and I. W.-H. Tsang. 2004. “The Pre-Image Problem in Kernel Methods.” IEEE Transactions on Neural Networks 15 (6): 1517–25. https://doi.org/10.1109/TNN.2004.837781.

Le, Quoc, Tamás Sarlós, and Alex Smola. 2013. “Fastfood-Approximating Kernel Expansions in Loglinear Time.” In Proceedings of the International Conference on Machine Learning. http://www.jmlr.org/proceedings/papers/v28/le13-supp.pdf.

Minh, Ha Quang, Partha Niyogi, and Yuan Yao. 2006. “Mercer’s Theorem, Feature Maps, and Smoothing.” In International Conference on Computational Learning Theory, 154–68. Lecture Notes in Computer Science. Springer. https://doi.org/10.1007/11776420_14.

Pourkamali-Anaraki, Farhad, and Stephen Becker. 2016a. “A Randomized Approach to Efficient Kernel Clustering,” August. http://arxiv.org/abs/1608.07597.

———. 2016b. “Randomized Clustered Nystrom for Large-Scale Kernel Machines,” December. http://arxiv.org/abs/1612.06470.

Rahimi, Ali, and Benjamin Recht. 2007. “Random Features for Large-Scale Kernel Machines.” In Advances in Neural Information Processing Systems, 1177–84. Curran Associates, Inc. http://papers.nips.cc/paper/3182-random-features-for-large-scale-kernel-machines.

———. 2009. “Weighted Sums of Random Kitchen Sinks: Replacing Minimization with Randomization in Learning.” In Advances in Neural Information Processing Systems, 1313–20. Curran Associates, Inc. http://papers.nips.cc/paper/3495-weighted-sums-of-random-kitchen-sinks-replacing-minimization-with-randomization-in-learning.

Scardapane, Simone, and Dianhui Wang. 2017. “Randomness in Neural Networks: An Overview.” Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery 7 (2). https://doi.org/10.1002/widm.1200.

Schölkopf, Bernhard, Phil Knirsch, Alex Smola, and Chris Burges. 1998. “Fast Approximation of Support Vector Kernel Expansions, and an Interpretation of Clustering as Approximation in Feature Spaces.” In Mustererkennung 1998, edited by Paul Levi, Michael Schanz, Rolf-Jürgen Ahlers, and Franz May, 125–32. Informatik Aktuell. Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-72282-0_12.

Schölkopf, Bernhard, and Alexander J. Smola. 2002. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press.

Schölkopf, Bernhard, Alexander Smola, and Klaus-Robert Müller. 1997. “Kernel Principal Component Analysis.” In Artificial Neural Networks — ICANN’97, edited by Wulfram Gerstner, Alain Germond, Martin Hasler, and Jean-Daniel Nicoud, 583–88. Lecture Notes in Computer Science. Springer Berlin Heidelberg. https://doi.org/10.1007/BFb0020217.

Strobl, Eric V., Kun Zhang, and Shyam Visweswaran. 2017. “Approximate Kernel-Based Conditional Independence Tests for Fast Non-Parametric Causal Discovery,” February. http://arxiv.org/abs/1702.03877.

Vedaldi, A., and A. Zisserman. 2012. “Efficient Additive Kernels via Explicit Feature Maps.” IEEE Transactions on Pattern Analysis and Machine Intelligence 34 (3): 480–92. https://doi.org/10.1109/TPAMI.2011.153.

Williams, Christopher K. I. 2001. “On a Connection Between Kernel PCA and Metric Multidimensional Scaling.” In Advances in Neural Information Processing Systems 13, edited by T. K. Leen, T. G. Dietterich, and V. Tresp, 46:675–81. MIT Press. https://doi.org/10.1023/A:1012485807823.

Williams, Christopher KI, and Matthias Seeger. 2001. “Using the Nyström Method to Speed up Kernel Machines.” In Advances in Neural Information Processing Systems, 682–88. http://papers.nips.cc/paper/1866-using-the-nystrom-method-to-speed-up-kernel-machines.

Yang, Jiyan, Vikas Sindhwani, Haim Avron, and Michael Mahoney. 2014. “Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels,” December. http://arxiv.org/abs/1412.8293.

Yang, Tianbao, Yu-Feng Li, Mehrdad Mahdavi, Rong Jin, and Zhi-Hua Zhou. 2012. “Nyström Method Vs Random Fourier Features: A Theoretical and Empirical Comparison.” In Advances in Neural Information Processing Systems, 476–84. http://papers.nips.cc/paper/4588-nystrom-method-vs-random-fourier-features-a-theoretical-and-empirical-comparison.

Yu, Chenhan D., William B. March, and George Biros. 2017. “An $N \Log N$ Parallel Fast Direct Solver for Kernel Matrices.” In. http://arxiv.org/abs/1701.02324.

Yu, Yaoliang, Hao Cheng, Dale Schuurmans, and Csaba Szepesvári. 2013. “Characterizing the Representer Theorem.” In Proceedings of the 30th International Conference on Machine Learning (ICML-13), 570–78. http://www.jmlr.org/proceedings/papers/v28/yu13.pdf.

Zhang, Jaslyn. 2017. “Improved Genomic Selection Using Vowpal Wabbit with Random Fourier Features.” https://dukespace.lib.duke.edu/dspace/handle/10161/14308.

Zhang, Qinyi, Sarah Filippi, Arthur Gretton, and Dino Sejdinovic. 2016. “Large-Scale Kernel Methods for Independence Testing,” June. http://arxiv.org/abs/1606.07892.