A straight line in a higher dimensional space can be a curvy line when projected onto a lower dimensional space. This straight line looks like a curvy line when we bring it down to the lower dimensional space. While there are many kernel approximation techniques to do the Kernel Trick, one prominent one is using the RBF Kernel.
We will also analyze two other Kernel approximation techniques, namely Nystrom Method and Random Features Technique, in this paper. Also, we would need to store all kernel values. This will only yield an approximate embedding but if we keep the number of samples we use the same, the resulting embedding will be independent of dataset size and we can basically choose the complexity to suit our problem. Random feature based methods use an element-wise approximation of the kernel. For the exact and Nystrom experiments, they apply a linear kernel.
For random features, they apply a hash kernel using MurmurHash3 as their hash function. Since they were predicting ratings for a review, they measured accuracy by using the root mean square error RMSE of the predicted rating as compared to the actual rating.
Ch 1 Support Vector Machine Solvers. Smola Eds. INFO E-mail: info videolectures. In this monograph Tanya Reinhart discusses strategies enabling the interface of different cognitive systems, which Musicant 2 1.
The dataset contained , training images and features per image. The authors started with these features in the dataset as input and used the RBF kernel for the exact and Nystrom method and random cosines for the random features method. The little black stars denote the end of an epoch For example, the hash random feature used for the Yelp dataset is much cheaper to compute than the string kernel. However, computing a block of the RBF kernel is similar in cost to computing a block of random cosine features.
Scalability of RBF Kernel Generation Figure: Time taken to compute on eblock of RBF kernel as they scale the number of examples and the number of machines used Here, ideal scaling implies that the time to generate a block of the kernel matrix remains constant as they increase both the data and the number of machines. However, computing a block of the RBF kernel involves broadcasting a b x d matrix to all the machines in the cluster.
This causes a slight decrease in performance as they go from 8 to machines. Large-scale Kernel Machines.
Chi-Jen Lin. Ch 12 Trading Convexity for Scalability.
A training algorithm for optimal margin classifiers. Duda and Peter E. Constant depth circuits, Fourier transform, and learnability. Creating surfaces from scattered data using radial basis functions. Daehlen, T. Lyche, and L. Burges, C. Atutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, , Cherkassky, V.
Learning from data-concepts, theory and methods. Cortes, C.
Support vector networks. Machine Learning, 20 , Dantzig, G. Decomposition principle for linear programs. Operations Research, 8 , Data for evaluating learning in valid experiments.
Gilmore, P. A linear programming approach to the cutting stock problem. Operations Research, 9 , Huber, P. Robust estimation of location parameter. Annals of Mathematical Statistics, 35 , Robust statistics. New York: John Wiley. Mangasarian, O. Generalized support vector machines. Smola, P.
Bartlett, B. Schuurmans Eds. Advances in large margin classifiers pp. Nonlinear perturbation of linear programs.