Abstract
The projection twin support vector machine (PTSVM) is a potential tool for classification problem. However the loss function of PTSVM is hinge loss function which is a unbounded loss and not robust enough to outliers. In this work, a robust PTSVM (termed RSHPTSVM) is proposed based on rescaled square hinge loss (RSH-loss) to handle classification problem. A close relationship between RSH-loss and correntropy is established theoretically. The RSH-loss can be viewed as a correntropy-induced loss by a reproducing piecewise kernel. As such a correntropy loss, it has vastly different properties from hinge loss such as boundedness, robustness and nonconvexity. Moreover, RSH-loss is with higher order statistical information from samples. However the nonconvexity of RSHPTSVM makes it difficult to optimize, so that an efficient iterative optimization algorithm based on semi-quadratic optimization theory is proposed to solve RSHPTSVM, which can quickly converge to the optimal solution. Furthermore, we extend our RSHPTSVM from binary classification to multi-classification and propose a robust projection multi-birth support vector machine model (termed RSHPMBSVM). The proposed method is implemented on various datasets including three artificial datasets, UCI datasets, and a practical application dataset. The experiment results under no noise and label noise circumstance confirm the feasibility and effectiveness of the proposed methods.
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This work is supported by the National Nature Science Foundation of China (No.11471010 and No.11271367).
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Appendix A:: Semi-quadratic optimization
Appendix A:: Semi-quadratic optimization
The semi-quadratic optimization technique is an optimization method based on the conjugate function theory, which can solve the non-convex optimization problem. The main idea is to convert the original non-convex objective function into a semi-quadratic augmented objective function by introducing an auxiliary variable. Finally, the algorithm of alternating iteration is used to obtain the approximate solution to achieve the final solution of the original problem.
Definition 1
When a function F(α, β) satisfies the following conditions, we call it a semiquadratic function. That is, function F(α, β) is a convex function with respect to β when α is fixed, and F(α, β) is a quadratic function with respect to α when β is fixed.
Solution of semiquadratic function
For the following type of minimum problem:
where α = [α1, α2, … , αN]T ∈ RN, J(α) represents a convex penalty function, L(α) represents the loss function and satisfies \(L(\alpha ) = {\sum }_{i=1}^{N}L(\alpha _{i})\).
For the loss function L(α), fixed the original α, we use the semi-quadratic optimization technique to introduce a new auxiliary variable β = [β1, β2, … , βN]T ∈ RN so that the following expression holds:
where F(αi, βi) is a semi-quadratic function.
With expression (A.2), formulation (A.1) can be written as follows:
where \(F(\alpha ,\beta )={\sum }_{i=1}^{N}F(\alpha _{i},\beta _{i})\), \(G(\beta )={\sum }_{i=1}^{N}G(\beta _{i})\). This is a semi-quadratic optimization problem. We call the objected function in (A.3) as an augmented objective function. The specific derivation process is as follows:
When the following theorem holds, the semi-quadratic function F(α, β) in the augmented objective function can be written as \(F(\alpha ,\beta )={\sum }_{i=1}^{N}F_{M}(\alpha _{i},\beta _{i})\), where \(F_{M}(\alpha _{i},\beta _{i})=\frac {1}{2}\alpha _{i}{\beta _{i}^{2}}\).
Theorem 1
If the loss function L(.) satisfies the following conditions:
-
(1)
L(α) ≥ 0, and L(0) = 0;
-
(2)
L(α) = L(−α),∀α ∈ R;
-
(3)
\(L^{\prime }(\alpha ) \ge 0,\forall \alpha \ge 0\);
-
(4)
\(\forall \alpha \in R^{+},\exists L^{\prime \prime }(\alpha ),\) and \(L^{\prime \prime }(0^{+})\ge 0\);
-
(5)
\(L(\sqrt {\alpha })\) is the concave function on R+; then there must be a convex function G(α) that satisfies the following conditions:
$$ L(\alpha) = \inf_{\beta > 0} \left\{\frac{1}{2}\beta \alpha^{2}+G(\beta)\right\} $$(A.4)when α is taken, there is a minimum solution β∗ ≥ 0 on the right side of the above equation, which satisfies the following conditions:
$$ \inf_{\beta>0}\left\{ \frac{1}{2}\beta \alpha^{2} + G(\beta) \right\} = \frac{1}{2}\beta^{*}\alpha^{2} + G(\beta^{*}) $$(A.5)where the specific expression of β is as follows:
$$ \beta^{*}=\left\{ \begin{array}{lll} \frac{L^{\prime}(\alpha)}{\alpha} \qquad& \alpha > 0 \\ L^{\prime\prime}(0^{+}) \qquad & \alpha = 0 \\ \frac{L^{\prime}(-\alpha)}{-\alpha} \qquad & \alpha <0 \end{array} \right. $$(A.6)The proof of the above theorem can be found in [25].
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Ren, Q., Yang, L. A robust projection twin support vector machine with a generalized correntropy-based loss. Appl Intell 52, 2154–2170 (2022). https://doi.org/10.1007/s10489-021-02480-6
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DOI: https://doi.org/10.1007/s10489-021-02480-6