A. Smola, Regression Estimation with Support Vector Learning Machines, Technische University Mnchen, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....recognition ( 55] In many of these areas, SVMs have shown to out perform well established methods such as Neural Networks and Radial Basis Functions. This thesis only considers the use of SVMs for pattern classification, although they have also been applied to other areas such as Regression ([56]) and Novelty detection ( 58] In classification, SVMs are binary classifiers. They are used to build a decision boundary by mapping data from the original input space to a higher dimensional feature space (see figure 1.2) where the data can be separated using a linear hyperplane. Figure 1.2: ....

A. J. Smola. Regression estimation with support vector learning machines. Master 's thesis, Technische Universitat Munchen, 1996.

.... Applications of support vector machine classifiers include isolated handwritten digit recognition [46, 144, 145, 34] object recognition [12] face detection [127] and text categorization [82] The support vector machine has been extended beyond classification tasks to regression problems [149, 159, 54] and principle component analysis (PCA) 143, 146] 2.2.1 SVM Mathematical Programs The support vector machine classifier is obtained by solving an optimization problem with an objective function which balances a term forcing separation between A and B 45 and a term maximizing the margin of ....

A. Smola. Regression estimation with support vector learning machines. Master's thesis, Technische Universitat Munchen, 1996.

....given by (3) Proposition 2. 2) This explicit convex quadratic programming formulation in the original primal space of the problem is rather simple and easily interpretable, but does not seem to have been given previously in other works that have considered dual formulations of this problem [13] [29], 31] By using parametric perturbation results of linear programming [17] we show that, for all values of the parameter for some , a Huber linear estimator is just an ordinary least squares estimate (Proposition 2.3) On the other hand, for all sufficiently small values of , the ....

....1 norm solution (Proposition 2.4) Li and Swetits [13] have studied the dual (17) of the Huber M estimator quadratic program (9) They show that (17) is a least 2 norm formulation of the dual of the least 1 norm estimator. In addition, they give perturbation results for the solution of (17) Smola [29] also presents a dual formulation of the Huber M estimator quadratic program [31] In Section 3, we set up a convex quadratic program (23) for a nonlinear generalized support vector machine [33] 16] which extends the Huber loss function to large classes of nonlinear regression problems. Related ....

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A. Smola, Regression Estimation with Support Vector Learning Machines, master's thesis, Technische Universita t Mu nchen, Mu nchen, Germany, 1996.

.... Structural Risk Minimization (Shawe Taylor et al. 1996; Cristianini et al. 1998) The resulting algorithm is similar to Support Vector Machines (SVM) Cortes and Vapnik 1995; Vapnik 1995) which became popular in recent years due to their good generalization properties (Cortes and Vapnik 1995; Smola 1996; Scholkopf 1997; Girosi 1997; Wahba 1997; Weston et al. 1997; Pontil and Verri 1998; Weston and Watkins 1998) Since during learning and application of SVM s only inner products of object representations x i and x j have to be computed, the method of potential functions (the so called kernel ....

Smola, A. (1996). Regression estimation with Support Vector Learning Machines. Master's thesis, Technical University Munich.

....the solution implicitly contains support vectors that provide a description of the significant data for classification. Image Speech and Intelligent Systems Group 5 Support Vector Regression SVMs can also be applied to regression problems by the introduction of an alternative loss function, [17]. The loss function must be modified to include a distance measure. Figure 21 illustrates four possible loss functions. a) Quadratic (b) Least Modulus (c) Huber (d) e Insensitive Figure 21 Loss Functions The loss function in Figure 21(a) corresponds to the conventional least squares error ....

A. Smola. 1996. Regression Estimation with Support Vector Learning Machines. Technische Universität München.

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A. J. Smola. Regression estimation with support vector learning machines. Master's thesis, Fakultat fur Physik, Technische Universitat Munchen, Munich, 1996.

....is dense. This is not very surprising since in cases where much data is needed to model the hypothesis precisely, by definition one simply cannot achieve the same result with less data. However, sparsity ratios generally decrease as the volume of data increases (Figure 8) reaching a lower limit (Smola, 1996). Thus the method should be particularly useful for large data sets. Despite the fact that you can initiate the algorithm with the previous ff i solution every time a new data point is learnt, training times could still be too long as more data is learnt for a large data set. In this case one ....

Smola, A.J. (1996). Regression estimation with support vector learning machines. Master's thesis, Department of Physics, Technische Universitat Munchen.

....the data is dense. This is not very surprising since in cases where much data is needed to model the hypothesis precisely, by definition one simply cannot achieve the same result with less data. However, sparsity ratios generally decrease as the volume of data increases reaching a lower limit [Smola, 1996]. Thus the method should be particularly useful for large datasets. Finally, from Figures 5 and 7 we notice that instance selection appears to be a useful strategy for highlighting erroneous or critical samples. This could be exploited in the context of data cleaning in which potential outliers ....

A. J. Smola. Regression estimation with support vector learning machines. Master's thesis, Technische Universitat Munchen, 1996.

....3 A more direct way to see this, e.g. for # 0, is to consider nonnegative functions f(x) that have support only for kxk p #=OE. In this case the integrand is negative on it s support and thus also the integral itself. Cost Functions 9 is an admissible SV kernel. Moreover one can show [Smola, 1996, Vapnik et al. 1997] that k(x; x 0 ) B 2n 1 (kx Gamma x 0 k) with B k ( Delta) k O i=1 1 [ Gamma1=2;1=2] Delta) 30) B splines of order 2n 1, defined by the 2n 1 convolution of the unit inverval, are also admissible. Finally, if the input dimensionality is too low, one could ....

A. J. Smola. Regression estimation with support vector learning machines. Master's thesis, Technische Universitat Munchen, 1996.

....Estimation. The advantage of this setting is that in the low noise case, it generates sparse decompositions of f(x) in terms of the training data, i.e. in terms of Support Vectors. This advantage however vanishes for noisy data as the number of Support Vectors increases with the noise (see [31] for details) Unfortunately, independent of the noise level, the choice of a different regularization prevents such an efficient calculation scheme due to equation (45) as D Gamma1 K generally may not be assumed to be diagonal. Consequently, the expansion of f is only sparse in terms of fi ....

A. J. Smola. Regression estimation with support vector learning machines. Master's thesis, Technische Universitat Munchen, 1996.

....Estimation. The advantage of this setting is that in the low noise case, it generates sparse decompositions of f(x) in terms of the training data, i.e. in terms of Support Vectors. This advantage however vanishes for noisy data as the number of Support Vectors increases with the noise (see [36] for details) Unfortunately, independent of the noise level, the choice of a different regularization prevents such an efficient calculation scheme due to equation (35) as D Gamma1 K generally may not be assumed to be diagonal. Consequently, the expansion of f is only sparse in terms of fi ....

A. J. Smola. Regression estimation with support vector learning machines. Master's thesis, Technische Universitat Munchen, Fakultat fur Physik, 1996.

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A. Smola, Regression Estimation with Support Vector Learning Machines, Technische University Mnchen, 1996.

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