Choosing Multiple Parameters for Support Vector Machines
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3,607 citations
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...Finally, Section V summarizes the observations and concluding remarks to complete this paper....
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1,762 citations
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..., scaling coefficient) on each feature by using an alternating optimization procedure (Weston et al., 2001; Chapelle et al., 2002; Grandvalet and Canu, 2003)....
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...These derivatives can be used to optimize the individual parameters (e.g., scaling coefficient) on each feature by using an alternating optimization procedure (Weston et al., 2001; Chapelle et al., 2002; Grandvalet and Canu, 2003)....
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...Chapelle et al. (2002) calculate the derivative of the margin and the derivative of the radius (of the smallest sphere enclosing the training points) with respect to a kernel parameter, θ:...
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...Chapelle et al. (2002) calculate the derivative of the margin and the derivative of the radius (of the smallest sphere enclosing the training points) with respect to a kernel parameter, θ: ∂ ‖w‖22 ∂θ = − N∑ i=1 N∑ j=1 αiαjyiyj ∂k(xi,xj) ∂θ ∂R2 ∂θ = N∑ i=1 βi ∂k(xi,xi) ∂θ − N∑ i=1 N∑ j=1 βiβj…...
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1,625 citations
Cites methods from "Choosing Multiple Parameters for Su..."
...While this so-called “multiple kernel learning” problem can in principle be solved via cross-validation, several recent papers have focused on more efficient methods for kernel learning (Chapelle et al., 2002; Grandvalet & Canu, 2003; Lanckriet et al., 2004; Ong et al., 2003)....
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1,578 citations
Cites methods from "Choosing Multiple Parameters for Su..."
...Once we have mined a set of discriminative actionlets, a multiple kernel learning [4] approach is employed to learn an actionlet ensemble structure that combines these discriminative actionlets....
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1,532 citations
Cites background or methods from "Choosing Multiple Parameters for Su..."
...…cross-validationprovides an almost unbiased estimate of the true generalisation performance (Luntz and Brailovsky, 1969), and this is often cited as being an advantageous property of the leave-one-out estimator in the setting of model selection (e.g., Vapnik, 1998; Chapelle et al., 2002)....
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...The analytic leave-one-out cross-validation procedure described here can easily be adapted to form the basis of an efficient model selection strategy (cf. Chapelle et al., 2002; Cawley and Talbot, 2003; Bo et al., 2006)....
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..., 2001) and has been widely adopted (e.g., Mika et al., 1999; Weston, 1999; Billings and Lee, 2002; Chapelle et al., 2002; Chu et al., 2003; Stewart, 2003; Mika et al., 2003; Gold et al., 2005; Peña Centeno and D., 2006; Andelić et al., 2006; An et al., 2007; Chen et al., 2009)....
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...…this suite of benchmark data sets (Rätsch et al., 2001) and has been widely adopted (e.g., Mika et al., 1999; Weston, 1999; Billings and Lee, 2002; Chapelle et al., 2002; Chu et al., 2003; Stewart, 2003; Mika et al., 2003; Gold et al., 2005; Peña Centeno and D., 2006; Andelić et al., 2006;…...
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...It is straightforward to demonstrate that leave-one-out cross-validation provides an almost unbiased estimate of the true generalisation performance (Luntz and Brailovsky, 1969), and this is often cited as being an advantageous property of the leave-one-out estimator in the setting of model selection (e.g., Vapnik, 1998; Chapelle et al., 2002)....
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References
40,147 citations
"Choosing Multiple Parameters for Su..." refers background in this paper
...bound each term in the sum by 1 which gives the following bound on the number of errors made by the leave-one-out procedure ( Vapnik, 1995 ):...
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37,861 citations
"Choosing Multiple Parameters for Su..." refers background in this paper
...It can be shown that soft margin SVMs with quadratic penalization of errors can be considered as a special case of the hard margin version with the modi ed kernel [4, 6]...
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...Dealing with non-separability For the non-separable case, one needs to allow training errors which results in the so called soft margin SVM algorithm [4]....
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26,531 citations
"Choosing Multiple Parameters for Su..." refers background or methods in this paper
...Indeed, it can be shown [19] that 1 2 kwk(2) =W ( ); and the lemma can be applied to the standard SVM optimization problem (2), giving @kwk2 @ p = X̀...
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...For SVMs without threshold and with no training errors, Vapnik [19] proposed the following upper bound on the number of errors of the leave-one-out procedure: T = 1 ` R2 2 : where R and are the radius and the margin as de ned in theorem 1....
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...We introduce some standard notations for SVMs; for a complete description, see [19]....
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...Vapnik and Chapelle [20, 3] derived an estimate using the concept of span of support vectors....
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...4 Radius-margin bound For SVMs without threshold and with no training errors, Vapnik [19] proposed the following upper bound on the number of errors of the leave-one-out procedure: T = 1 ` R2 2 :...
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13,736 citations
12,530 citations
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...The second leukemia classi cation problem was discriminating B versus T cells for lymphoblastic cells [7]....
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...2 DNA Microarray Data Next, we tested this idea on two leukemia discrimination problems [7] and a problem of predicting treatment outcome for Medulloblastoma 6....
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