Duality and Geometry in SVM Classifiers

By Kristin P. Bennett, Erin J. Bredensteiner. 2000

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Citations in the corpus

On Security and Sparsity of Linear Classifiers for Adversarial Settings (2016, methodology): “In fact, in the presence of sparse attacks, Infinity-norm SVMs can drastically outperform the security of standard SVMs.”
Support vector machine classification with indefinite kernels (2007, methodology): “(See Bennet and Bredensteiner (2000) for a discussion on geometric interpretations in SVM.) Another direction applies direct spectral transformations to indefinite kernels: flipping the negative eigenvalues or shifting the eigenvalues and reconstructing the kernel with the original eigenvectors in o”
A tutorial on?-support vector machines (2005, background): “For further details on the connection between ν-SVMs and C-SVMs, see [16, 6].”
A Robust Minimax Approach to Classification (2002, result): “The optimal worst-case misclassification probability is related to the above optimal value κ An interesting analogy to this result can be found in SVM classification (Bennett and Bredensteiner , 2000; Crisp and Burges, 1999).”
A tutorial on support vector regression (2004, unknown): “(6) Note that by α(∗) i , we refer to αi and α ∗ i .”
Support Vector Machine Solvers (2007, background): “There are also alternate formulations of the dual problem (Keerthi et al., 1999; Bennett and Bredensteiner, 2000).”
Materials that couple sensing, actuation, computation, and communication (2015, background): “The separating hyperplane can then be reduced to finding the closest points between the two convex hulls, and selecting a hyperplane which bisects and is orthogonal to the line connecting these two points [14].”
Boosting Classifiers with Tightened L0-Relaxation Penalties (2010, background): “…is to maximize the margin of separation, minus a misclassification penalty; however, this formulation is known to be equivalent, for some appropriate choice of the constant D′, to min λ,ξ≥0 { U∑ u=1 λu + D′ M∑ i=1 ξi | diag(y)Hλ + ξ ≥ 1 } ; (4) see for example Bennett & Bredensteiner (2000).”
Coresets for polytope distance (2009, background): “(C-SVM) yi(w φ(xi)− b) ≥ 1− ξi, ξi≥0 ∀i [7] 2d two-class `1-SVM min w,b,ρ,ξ 1 2 ||w||(2) − ρ+ ν 2 P i ξi two (reduced) polytopes”
Learning the Kernel Matrix with Semidefinite Programming (2002, background): “Geometrically, γ corresponds to the distance between the convex hulls (the smallest convex sets that contain the data in each class) of the two classes (Bennett and Bredensteiner, 2000).”
Convex sets as prototypes for classifying patterns (2009, methodology): “Recently, convex hulls spanned by the training points of each class have been of interest in the machine learning field (Bennett and Bredensteiner, 2000; Zhou et al., 2004; Mavroforakis et al., 2006) since those convex hulls can examine the separability among the given classes and they are also…”
Distance-Based Classification with Lipschitz Functions (2004, methodology): “In Bennett and Bredensteiner (2000) and Zhou et al. (2002) it has been established that constructing a maximal margin hyperplane between the set X+ of positive and X− of negative training points in a Banach space V is equivalent to finding the distance between the convex hulls of X+ and X−.”
A novel Frank-Wolfe algorithm. Analysis and applications to large-scale SVM training (2013, methodology): “In a closely related work [38], Kumar and Yildirim present a specialization of the MFW method to SVM problems, adopting the geometrical formulation studied in [6].”