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238
Pegasos: Primal Estimated subgradient solver for SVM
"... We describe and analyze a simple and effective stochastic subgradient descent algorithm for solving the optimization problem cast by Support Vector Machines (SVM). We prove that the number of iterations required to obtain a solution of accuracy ɛ is Õ(1/ɛ), where each iteration operates on a singl ..."
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Cited by 541 (21 self)
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We describe and analyze a simple and effective stochastic subgradient descent algorithm for solving the optimization problem cast by Support Vector Machines (SVM). We prove that the number of iterations required to obtain a solution of accuracy ɛ is Õ(1/ɛ), where each iteration operates on a single training example. In contrast, previous analyses of stochastic gradient descent methods for SVMs require Ω(1/ɛ2) iterations. As in previously devised SVM solvers, the number of iterations also scales linearly with 1/λ, where λ is the regularization parameter of SVM. For a linear kernel, the total runtime of our method is Õ(d/(λɛ)), where d is a bound on the number of nonzero features in each example. Since the runtime does not depend directly on the size of the training set, the resulting algorithm is especially suited for learning from large datasets. Our approach also extends to nonlinear kernels while working solely on the primal objective function, though in this case the runtime does depend linearly on the training set size. Our algorithm is particularly well suited for large text classification problems, where we demonstrate an orderofmagnitude speedup over previous SVM learning methods.
Using the Nyström Method to Speed Up Kernel Machines
 Advances in Neural Information Processing Systems 13
, 2001
"... A major problem for kernelbased predictors (such as Support Vector Machines and Gaussian processes) is that the amount of computation required to find the solution scales as O(n ), where n is the number of training examples. We show that an approximation to the eigendecomposition of the Gram matrix ..."
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Cited by 432 (6 self)
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A major problem for kernelbased predictors (such as Support Vector Machines and Gaussian processes) is that the amount of computation required to find the solution scales as O(n ), 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 Nyström 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(m n). 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.
Efficient Additive Kernels via Explicit Feature Maps
"... Maji and Berg [13] have recently introduced an explicit feature map approximating the intersection kernel. This enables efficient learning methods for linear kernels to be applied to the nonlinear intersection kernel, expanding the applicability of this model to much larger problems. In this paper ..."
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Cited by 244 (9 self)
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Maji and Berg [13] have recently introduced an explicit feature map approximating the intersection kernel. This enables efficient learning methods for linear kernels to be applied to the nonlinear intersection kernel, expanding the applicability of this model to much larger problems. In this paper we generalize this idea, and analyse a large family of additive kernels, called homogeneous, in a unified framework. The family includes the intersection, Hellinger’s, and χ2 kernels commonly employed in computer vision. Using the framework we are able to: (i) provide explicit feature maps for all homogeneous additive kernels along with closed form expression for all common kernels; (ii) derive corresponding approximate finitedimensional feature maps based on the Fourier sampling theorem; and (iii) quantify the extent of the approximation. We demonstrate that the approximations have indistinguishable performance from the full kernel on a number of standard datasets, yet greatly reduce the train/test times of SVM implementations. We show that the χ2 kernel, which has been found to yield the best performance in most applications, also has the most compact feature representation. Given these train/test advantages we are able to obtain a significant performance improvement over current state of the art results based on the intersection kernel. 1.
On the Nyström Method for Approximating a Gram Matrix for Improved KernelBased Learning
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2005
"... A problem for many kernelbased 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 easilyinterpretable lowrank approximation to an nn Gram matrix G such that compu ..."
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Cited by 188 (11 self)
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A problem for many kernelbased 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 easilyinterpretable lowrank approximation to an nn Gram matrix G such that computations of interest may be performed more rapidly. The approximation is of the form G k = CW , where C is a matrix consisting of a small number c of columns of G and W k is the best rankk 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 judiciouslychosen and datadependent nonuniform probability distribution. Let F denote the spectral norm and the Frobenius norm, respectively, of a matrix, and let G k be the best rankk approximation to G. We prove that by choosing O(k/# ) columns both in expectation and with high probability, for both # = 2, F , and for all k : 0 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 Nyström method from integral equation theory are discussed.
Training a support vector machine in the primal
 Neural Computation
, 2007
"... Most literature on Support Vector Machines (SVMs) concentrate on the dual optimization problem. In this paper, we would like to point out that the primal problem can also be solved efficiently, both for linear and nonlinear SVMs, and that there is no reason for ignoring this possibilty. On the cont ..."
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Cited by 155 (5 self)
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Most literature on Support Vector Machines (SVMs) concentrate on the dual optimization problem. In this paper, we would like to point out that the primal problem can also be solved efficiently, both for linear and nonlinear SVMs, and that there is no reason for ignoring this possibilty. On the contrary, from the primal point of view new families of algorithms for large scale SVM training can be investigated.
The Kernel Recursive Least Squares Algorithm
 IEEE Transactions on Signal Processing
, 2003
"... We present a nonlinear kernelbased version of the Recursive Least Squares (RLS) algorithm. Our KernelRLS (KRLS) algorithm performs linear regression in the feature space induced by a Mercer kernel, and can therefore be used to recursively construct the minimum mean squared error regressor. Spars ..."
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Cited by 141 (2 self)
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We present a nonlinear kernelbased version of the Recursive Least Squares (RLS) algorithm. Our KernelRLS (KRLS) algorithm performs linear regression in the feature space induced by a Mercer kernel, and can therefore be used to recursively construct the minimum mean squared error regressor. Sparsity of the solution is achieved by a sequential sparsification process that admits into the kernel representation a new input sample only if its feature space image cannot be suffciently well approximated by combining the images of previously admitted samples. This sparsification procedure is crucial to the operation of KRLS, as it allows it to operate online, and by effectively regularizing its solutions. A theoretical analysis of the sparsification method reveals its close affinity to kernel PCA, and a datadependent loss bound is presented, quantifying the generalization performance of the KRLS algorithm. We demonstrate the performance and scaling properties of KRLS and compare it to a stateof theart Support Vector Regression algorithm, using both synthetic and real data. We additionally test KRLS on two signal processing problems in which the use of traditional leastsquares methods is commonplace: Time series prediction and channel equalization.
Core vector machines: Fast SVM training on very large data sets
 Journal of Machine Learning Research
, 2005
"... Standard SVM training has O(m 3) time and O(m 2) space complexities, where m is the training set size. It is thus computationally infeasible on very large data sets. By observing that practical SVM implementations only approximate the optimal solution by an iterative strategy, we scale up kernel met ..."
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Cited by 137 (15 self)
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Standard SVM training has O(m 3) time and O(m 2) space complexities, where m is the training set size. It is thus computationally infeasible on very large data sets. By observing that practical SVM implementations only approximate the optimal solution by an iterative strategy, we scale up kernel methods by exploiting such “approximateness ” in this paper. We first show that many kernel methods can be equivalently formulated as minimum enclosing ball (MEB) problems in computational geometry. Then, by adopting an efficient approximate MEB algorithm, we obtain provably approximately optimal solutions with the idea of core sets. Our proposed Core Vector Machine (CVM) algorithm can be used with nonlinear kernels and has a time complexity that is linear in m and a space complexity that is independent of m. Experiments on large toy and realworld data sets demonstrate that the CVM is as accurate as existing SVM implementations, but is much faster and can handle much larger data sets than existing scaleup methods. For example, CVM with the Gaussian kernel produces superior results on the KDDCUP99 intrusion detection data, which has about five million training patterns, in only 1.4 seconds on a 3.2GHz Pentium–4 PC.
Sparse Greedy Gaussian Process Regression
 Advances in Neural Information Processing Systems 13
, 2001
"... We present a simple sparse greedy technique to approximate the maximum a posteriori estimate of Gaussian Processes with much improved scaling behaviour in the sample size m. In particular, computational requirements are O(n m), storage is O(nm), the cost for prediction is O(n) and the cost to comput ..."
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Cited by 130 (1 self)
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We present a simple sparse greedy technique to approximate the maximum a posteriori estimate of Gaussian Processes with much improved scaling behaviour in the sample size m. In particular, computational requirements are O(n m), storage is O(nm), the cost for prediction is O(n) and the cost to compute confidence bounds is O(nm), where n m. We show how to compute a stopping criterion, give bounds on the approximation error, and show applications to large scale problems.
Learning spectral clustering
, 2003
"... Spectral clustering refers to a class of techniques which rely on the eigenstructure of a similarity matrix to partition points into disjoint clusters with points in the same cluster having high similarity and points in different clusters having low similarity. In this paper, we derive a new cost fu ..."
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Cited by 119 (4 self)
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Spectral clustering refers to a class of techniques which rely on the eigenstructure of a similarity matrix to partition points into disjoint clusters with points in the same cluster having high similarity and points in different clusters having low similarity. In this paper, we derive a new cost function for spectral clustering based on a measure of error between a given partition and a solution of the spectral relaxation of a minimum normalized cut problem. Minimizing this cost function with respect to the partition leads to a new spectral clustering algorithm. Minimizing with respect to the similarity matrix leads to an algorithm for learning the similarity matrix. We develop a tractable approximation of our cost function that is based on the power method of computing eigenvectors. 1