text: This paper concerns the distributed training of nonlinear kernel machines on Map-Reduce. We show that a re-formulation of Nystr "om approximation based solution which is solved using gradient based techniques is well suited for this, especially when it is necessary to work with a large number of basis points. The main advantages of this approach are: avoidance of computing the pseudo-inverse of the kernel sub-matrix corresponding to the basis points; simplicity and efficiency of the distributed part of the computations; and, friendliness to stage-wise addition of basis points. We implement the method using an AllReduce tree on Hadoop and demonstrate its value on a few large benchmark datasets.
  @cite_26: 
  @cite_7: Low-rank matrix decomposition has gained great popularity recently in scaling up kernel methods to large amounts of data. However, some limitations could prevent them from working effectively in certain domains. For example, many existing approaches are intrinsically unsupervised, which does not incorporate side information (e.g., class labels) to produce task specific decompositions; also, they typically work "transductively", i.e., the factorization does not generalize to new samples, so the complete factorization needs to be recomputed when new samples become available. To solve these problems, in this paper we propose an "inductive"-flavored method for low-rank kernel decomposition with priors. We achieve this by generalizing the Nystrom method in a novel way. On the one hand, our approach employs a highly flexible, nonparametric structure that allows us to generalize the low-rank factors to arbitrarily new samples; on the other hand, it has linear time and space complexities, which can be orders of magnitudes faster than existing approaches and renders great efficiency in learning a low-rank kernel decomposition. Empirical results demonstrate the efficacy and efficiency of the proposed method.
  @cite_9: Support Vector Machines (SVMs) suffer from a widely recognized scalability problem in both memory use and computational time. To improve scalability, we have developed a parallel SVM algorithm (PSVM), which reduces memory use through performing a row-based, approximate matrix factorization, and which loads only essential data to each machine to perform parallel computation. Let n denote the number of training instances, p the reduced matrix dimension after factorization (p is significantly smaller than n), and m the number of machines. PSVM reduces the memory requirement from O(n2) to O(np m), and improves computation time to O(np2 m). Empirical study shows PSVM to be effective. PSVM Open Source is available for download at http: code.google.com p psvm .
  @cite_21: 
  @cite_6: A problem for many kernel-based methods is that the amount of computation required to find the solution scales as O(n3), where n is the number of training examples. We develop and analyze an algorithm to compute an easily-interpretable low-rank approximation to an n × n Gram matrix G such that computations of interest may be performed more rapidly. The approximation is of the form Gk = CWk+CT, where C is a matrix consisting of a small number c of columns of G and Wk is the best rank-k approximation to W, the matrix formed by the intersection between those c columns of G and the corresponding c rows of G. An important aspect of the algorithm is the probability distribution used to randomly sample the columns; we will use a judiciously-chosen and data-dependent nonuniform probability distribution. Let ||·||2 and ||·||F denote the spectral norm and the Frobenius norm, respectively, of a matrix, and let Gk be the best rank-k approximation to G. We prove that by choosing O(k e4) columns||G-CWk+CT||ξ ≤ ||G-Gk||ξ + e Σi=1n Gii2 ,both in expectation and with high probability, for both ξ = 2, F, and for all k: 0 ≤ k ≤ rank(W). This approximation can be computed using O(n) additional space and time, after making two passes over the data from external storage. The relationships between this algorithm, other related matrix decompositions, and the Nystrom method from integral equation theory are discussed.
  @cite_24: 
  @cite_27: Kernel (or similarity) matrix plays a key role in many machine learning algorithms such as kernel methods, manifold learning, and dimension reduction. However, the cost of storing and manipulating the complete kernel matrix makes it infeasible for large problems. The Nystrom method is a popular sampling-based low-rank approximation scheme for reducing the computational burdens in handling large kernel matrices. In this paper, we analyze how the approximating quality of the Nystrom method depends on the choice of landmark points, and in particular the encoding powers of the landmark points in summarizing the data. Our (non-probabilistic) error analysis justifies a “clustered Nystrom method” that uses the k-means clustering centers as landmark points. Our algorithm can be applied to scale up a wide variety of algorithms that depend on the eigenvalue decomposition of kernel matrix (or its variant), such as kernel principal component analysis, Laplacian eigenmap, spectral clustering, as well as those involving kernel matrix inverse such as least-squares support vector machine and Gaussian process regression. Extensive experiments demonstrate the competitive performance of our algorithm in both accuracy and efficiency.
  @cite_23: 
  @cite_25: A major problem for kernel-based predictors (such as Support Vector Machines and Gaussian processes) is that the amount of computation required to find the solution scales as O(n3), where n is the number of training examples. We show that an approximation to the eigendecomposition of the Gram matrix can be computed by the Nystrom method (which is used for the numerical solution of eigenproblems). This is achieved by carrying out an eigendecomposition on a smaller system of size m < n, and then expanding the results back up to n dimensions. The computational complexity of a predictor using this approximation is O(m2n). We report experiments on the USPS and abalone data sets and show that we can set m ≪ n without any significant decrease in the accuracy of the solution.
  @cite_17: 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.
summary: There are methods that use a low rank approximation of the kernel. @cite_9 and Woodsend & Gonzio @cite_26 use low rank Cholesky factorization of the kernel matrix. There is a large literature based on using Nystr "om method @cite_6 @cite_23 @cite_17 @cite_25 @cite_21 @cite_27 @cite_7 @cite_24 . While the methods in this class are useful for getting decent sub-optimal performance, they are insufficient if we aim to achieve a performance matching the true kernel machine.