Kernel principal component analysis (KPCA) provides a concise set of basis vectors which capture non-linear structures within large data sets, and is a central tool in data analysis and learning. To allow for non-linear relations, typically a full @math kernel matrix is constructed over @math data points, but this requires too much space and time for large values of @math . Techniques such as the Nystr "om method and random feature maps can help towards this goal, but they do not explicitly maintain the basis vectors in a stream and take more space than desired. We propose a new approach for streaming KPCA which maintains a small set of basis elements in a stream, requiring space only logarithmic in @math , and also improves the dependence on the error parameter. Our technique combines together random feature maps with recent advances in matrix sketching, it has guaranteed spectral norm error bounds with respect to the original kernel matrix, and it compares favorably in practice to state-of-the-art approaches.
In this line of work, the kernel matrix is approximated via randomized feature maps. The seminal work of @cite_8 showed one can construct randomized feature maps @math such that for any shift-invariant kernel @math and all @math , @math and if @math , then with probability at least @math , @math . Using this mapping, instead of lifting data points to @math by the kernel trick, they embed the data to a low-dimensional Euclidean inner product space. Subsequent works generalized to other kernel functions such as group invariant kernels @cite_21 , min intersection kernels @cite_17 , dot-product kernels @cite_16 , and polynomial kernels @cite_12 @cite_10 . This essentially converts kernel PCA to linear PCA. In particular, Lopez al @cite_15 proposed , which is an exact linear PCA on the approximate data feature maps matrix @math . They showed the approximation error is bounded as @math , where @math is not actually constructed.
@cite_8 @cite_21 @cite_15 @cite_16 @cite_10 @cite_12 @cite_17
1988813039 2243542436 2146897752 2158228027
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. Modern learning problems in computer vision, natural language processing, computational biology, and other areas are often based on large data sets of tens of thousands to millions of training instances. However, several standard learning algorithms, such as kernel-based algorithms, e.g., Support Vector Machines, Kernel Ridge Regression, Kernel PCA, do not easily scale to such orders of magnitude. This thesis focuses on sampling-based matrix approximation techniques that help scale kernel-based algorithms to large-scale datasets. We address several fundamental theoretical and empirical questions including: (1) What approximation should be used? We discuss two common sampling-based methods, providing novel theoretical insights regarding their suitability for various applications and experimental results motivated by this theory. Our results show that one of these methods, the Nystrom method, is superior in the context of large-scale learning. (2) Do these approximations work in practice? We show the effectiveness of approximation techniques on a variety of problems. In the largest study to-date for manifold learning, we use the Nystrom method to extract low-dimensional structure from high-dimensional data to effectively cluster face images. We also report good empirical results for Kernel Ridge Regression and Kernel Logistic Regression. (3) How should we sample columns? A key aspect of sampling-based algorithms is the distribution according to which columns are sampled. We study both fixed and adaptive sampling schemes as well as a promising ensemble technique that can be easily parallelized and generates superior approximations, both in theory and in practice. (4) How well do these approximations work in theory? We provide theoretical analyses of the Nystrom method to understand when this technique should be used. We present guarantees on approximation accuracy based on various matrix properties and analyze the effect of matrix approximation on actual kernel-based algorithms. This work has important consequences for the machine learning community since it extends to large-scale applications the benefits of kernel-based algorithms. The crucial aspect of this research, involving low-rank matrix approximation, is of independent interest within the field of linear algebra. 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. Kernel approximation using randomized feature maps has recently gained a lot of interest. In this work, we identify that previous approaches for polynomial kernel approximation create maps that are rank deficient, and therefore do not utilize the capacity of the projected feature space effectively. To address this challenge, we propose compact random feature maps (CRAFTMaps) to approximate polynomial kernels more concisely and accurately. We prove the error bounds of CRAFTMaps demonstrating their superior kernel reconstruction performance compared to the previous approximation schemes. We show how structured random matrices can be used to efficiently generate CRAFTMaps, and present a single-pass algorithm using CRAFTMaps to learn non-linear multi-class classifiers. We present experiments on multiple standard data-sets with performance competitive with state-of-the-art results.