The Nystr "om methods have been popular techniques for scalable kernel based learning. They approximate explicit, low-dimensional feature mappings for kernel functions from the pairwise comparisons with the training data. However, Nystr "om methods are generally applied without the supervision provided by the training labels in the classification regression problems. This leads to pairwise comparisons with randomly chosen training samples in the model. Conversely, this work studies a supervised Nystr "om method that chooses the critical subsets of samples for the success of the Machine Learning model. Particularly, we select the Nystr "om support vectors via the negative margin criterion, and create explicit feature maps that are more suitable for the classification task on the data. Experimental results on six datasets show that, without increasing the complexity over unsupervised techniques, our method can significantly improve the classification performance achieved via kernel approximation methods and reduce the number of features needed to reach or exceed the performance of the full-dimensional kernel machines.
Many data independent approximations of kernel based features have been proposed. Rahimi and Recht introduced random features to approximate shift invariant kernels @cite_37 @cite_26 . The most well-known of such techniques is the . These methods were later extended for improved complexity and versatility @cite_41 @cite_13 . Random features have desirable generalization properties @cite_38 , even though data dependent approximations were shown to exploit the structure in the data better, both theoretically and empirically @cite_2 @cite_38 .
@cite_38 @cite_37 @cite_26 @cite_41 @cite_2 @cite_13
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Approximations based on random Fourier features have recently emerged as an efficient and elegant methodology for designing large-scale kernel machines [4]. By expressing the kernel as a Fourier expansion, features are generated based on a finite set of random basis projections with inner products that are Monte Carlo approximations to the original kernel. However, the original Fourier features are only applicable to translation-invariant kernels and are not suitable for histograms that are always non-negative. This paper extends the concept of translation-invariance and the random Fourier feature methodology to arbitrary, locally compact Abelian groups. Based on empirical observations drawn from the exponentiated χ2 kernel, the state-of-the-art for histogram descriptors, we propose a new group called the skewed-multiplicative group and design translation-invariant kernels on it. Experiments show that the proposed kernels outperform other kernels that can be similarly approximated. In a semantic segmentation experiment on the PASCAL VOC 2009 dataset, the approximation allows us to train large-scale learning machines more than two orders of magnitude faster than previous nonlinear SVMs. With the goal of accelerating the training and testing complexity of nonlinear kernel methods, several recent papers have proposed explicit embeddings of the input data into low-dimensional feature spaces, where fast linear methods can instead be used to generate approximate solutions. Analogous to random Fourier feature maps to approximate shift-invariant kernels, such as the Gaussian kernel, on Rd, we develop a new randomized technique called random Laplace features, to approximate a family of kernel functions adapted to the semigroup structure of R+d. This is the natural algebraic structure on the set of histograms and other non-negative data representations. We provide theoretical results on the uniform convergence of random Laplace features. Empirical analyses on image classification and surveillance event detection tasks demonstrate the attractiveness of using random Laplace features relative to several other feature maps proposed in the literature. We consider the problem of improving the efficiency of randomized Fourier feature maps to accelerate training and testing speed of kernel methods on large datasets. These approximate feature maps arise as Monte Carlo approximations to integral representations of shift-invariant kernel functions (e.g., Gaussian kernel). In this paper, we propose to use Quasi-Monte Carlo (QMC) approximations instead, where the relevant integrands are evaluated on a low-discrepancy sequence of points as opposed to random point sets as in the Monte Carlo approach. We derive a new discrepancy measure called box discrepancy based on theoretical characterizations of the integration error with respect to a given sequence. We then propose to learn QMC sequences adapted to our setting based on explicit box discrepancy minimization. Our theoretical analyses are complemented with empirical results that demonstrate the effectiveness of classical and adaptive QMC techniques for this problem. Approximation of non-linear kernels using random feature mapping has been successfully employed in large-scale data analysis applications, accelerating the training of kernel machines. While previous random feature mappings run in O(ndD) time for @math training samples in d-dimensional space and D random feature maps, we propose a novel randomized tensor product technique, called Tensor Sketching, for approximating any polynomial kernel in O(n(d+D D )) time. Also, we introduce both absolute and relative error bounds for our approximation to guarantee the reliability of our estimation algorithm. Empirically, Tensor Sketching achieves higher accuracy and often runs orders of magnitude faster than the state-of-the-art approach for large-scale real-world datasets.