Sparse least squares support vector machine with L&lt;inf&gt;0&lt;/inf&gt;-norm in primal space

Li, Qi; Li, Xiaohang; Ba, Wei

doi:10.1109/icinfa.2015.7279758

Cited by 4 publications

(8 citation statements)

References 10 publications

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“…To this end, it should be noted that the definition (10) of the indicator function s(e) at e = 0 (either 0 or 1) does not influence the first-order differential of L(e) given in (11). It is demonstrated in the following that this definition has also no influence on the solutions of (12) and (16).…”

Section: The Matrixmentioning

confidence: 89%

“…Correspondingly, substituting (27), (29), (36), (37) and (38) into (16), the error vector defined in (14) is thus partitioned as follows…”

Section: Partition Of the Training Patternsmentioning

confidence: 99%

“…As discussed in section 3, solving the optimization problem (6) subject to the L2-SVM loss (7) is equivalent to solving the nonlinear equation (16) for S and e subject to (10). However, this equation is difficult to solve analytically,…”

Section: An Iterative Algorithmmentioning

confidence: 99%

“…This gives the strong impression that this is the only possible way to construct a SVM, particularly for SVMs with nonlinear kernels. There has aslo been quite a lot of interest in studying systems that have particular properties, for example [15]'s work on sparse learning-and [16]'s sparse least squares. However, there has been increasing interest in constructing SVMs directly in the primal.…”

Section: Introductionmentioning

confidence: 99%

“…For this generalization, it is assumed that: a) the full kernel matrix is symmetrical, and b) any principal minor of the full kernel matrix is positive semi-definite.These assumptions guarantee that the RLS problem has a unique optimal solution. In addition, matrix I + CPP T SB is invertible in(16) for any S when the full inner product matrix PP T is replaced in a general form diag(y) X, X diag(y) for nonlinear kernels. This will be discussed again later.…”

mentioning

confidence: 99%

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Building support vector machines in the context of regularized least squares

2016

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This paper formulates a linear kernel support vector machine (SVM) as a regularized least-squares (RLS) problem. By defining a set of indicator variables of the errors, the solution to the RLS problem is represented as an equation that relates the error vector to the indicator variables. Through partitioning the training set, the SVM weights and bias are expressed analytically using the support vectors. It is also shown how this approach naturally extends to Sums with nonlinear kernels whilst avoiding the need to make use of Lagrange multipliers and duality theory. A fast iterative solution algorithm based on Cholesky decomposition with permutation of the support vectors is suggested as a solution method. The properties of our SVM formulation are analyzed and compared with standard SVMs using a simple example that can be illustrated graphically. The correctness and behavior of our solution (merely derived in the primal context of RLS) is demonstrated using a set of public benchmarking problems for both linear and nonlinear SVMs.

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Section: The Matrixmentioning

confidence: 89%

“…Correspondingly, substituting (27), (29), (36), (37) and (38) into (16), the error vector defined in (14) is thus partitioned as follows…”

Section: Partition Of the Training Patternsmentioning

confidence: 99%

Section: An Iterative Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%