Kernel approximation via nonlinear random feature maps is widely used in speeding up kernel machines. There are two main challenges for the conventional kernel approximation methods. First, before performing kernel approximation, a good kernel has to be chosen. Picking a good kernel is a very challenging problem in itself. Second, high-dimensional maps are often required in order to achieve good performance. This leads to high computational cost in both generating the nonlinear maps, and in the subsequent learning and prediction process. In this work, we propose to optimize the nonlinear maps directly with respect to the classification objective in a data-dependent fashion. The proposed approach achieves kernel approximation and kernel learning in a joint framework. This leads to much more compact maps without hurting the performance. As a by-product, the same framework can also be used to achieve more compact kernel maps to approximate a known kernel. We also introduce Circulant Nonlinear Maps, which uses a circulant-structured projection matrix to speed up the nonlinear maps for high-dimensional data.
Following the seminal work on explicit nonlinear feature maps for approximating positive definite shift-invariant kernels @cite_10 , nonlinear mapping techniques have been proposed to approximate other forms of kernels such as the polynomial kernel @cite_19 @cite_28 , generalized RBF kernels @cite_9 , intersection kernels @cite_20 , additive kernels @cite_16 , skewed multiplicative histogram kernels @cite_29 , and semigroup kernel @cite_34 . Techniques have also been proposed to improve the speed and compactness of kernel approximations by using structured projections @cite_47 , better quasi Monte Carlo sampling @cite_21 , binary code @cite_2 @cite_31 , and dimensionality reduction @cite_25 . Our method in this paper is built upon the Random Fourier Feature @cite_10 for approximating shift-invariant kernel, a widely used kernel type in machine learning. Besides explicit nonlinear maps, kernel approximation can also be achieved using sampling-based low-rank approximations of the kernel matrices such as the Nystrom method @cite_1 @cite_49 @cite_12 . In order for these approximations to work well, the eigenspectrum of the kernel matrix should have a large gap @cite_5 .
@cite_31 @cite_28 @cite_9 @cite_29 @cite_21 @cite_1 @cite_34 @cite_19 @cite_5 @cite_49 @cite_2 @cite_12 @cite_47 @cite_16 @cite_10 @cite_25 @cite_20
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Kernel methods yield state-of-the-art performance in certain applications such as image classification and object detection. However, large scale problems require machine learning techniques of at most linear complexity and these are usually limited to linear kernels. This unfortunately rules out gold-standard kernels such as the generalized RBF kernels (e.g. exponential-c 2 ). Recently, Maji and Berg [13] and Vedaldi and Zisserman [20] proposed explicit feature maps to approximate the additive kernels (intersection, c 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 [14] 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. Furthermore, we investigate a learning method using l 1 regularization to encourage sparsity in the final vector representation, and thus reduce its dimension. We evaluate this technique on the VOC 2007 detection challenge, showing when it can improve on fast additive kernels, and the trade-offs in complexity and accuracy. 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. 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. Kernel normalization methods have been employed to improve robustness of optimization methods to reparametrization of convolution kernels, covariate shift, and to accelerate training of Convolutional Neural Networks (CNNs). However, our understanding of theoretical properties of these methods has lagged behind their success in applications. We develop a geometric framework to elucidate underlying mechanisms of a diverse range of kernel normalization methods. Our framework enables us to expound and identify geometry of space of normalized kernels. We analyze and delineate how state-of-the-art kernel normalization methods affect the geometry of search spaces of the stochastic gradient descent (SGD) algorithms in CNNs. Following our theoretical results, we propose a SGD algorithm with assurance of almost sure convergence of the methods to a solution at single minimum of classification loss of CNNs. Experimental results show that the proposed method achieves state-of-the-art performance for major image classification benchmarks with CNNs.