Regularization perspectives on support vector machines: Difference between revisions

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When <math>\mathcal{H}</math> is a [[reproducing kernel Hilbert space]], there exists a [[kernel function]] <math>K \colon \mathbf X \times \mathbf X \to \mathbb R</math> that can be written as an <math>n \times n</math> [[symmetric]] [[Positive-definite kernel|positive-definite]] [[matrix (mathematics)|matrix]] <math>\mathbf K</math>. By the [[representer theorem]],<ref>See {{cite book |last=Scholkopf |first=Bernhard |author2=Ralf Herbrich |author3=Alex Smola |title=A Generalized Representer Theorem |journal=Computational Learning Theory: Lecture Notes in Computer Science |year=2001 |volume=2111 |pages=416–426 |doi=10.1007/3-540-44581-1_27 |series=Lecture Notes in Computer Science |isbn=978-3-540-42343-0 |citeseerx=10.1.1.42.8617 }}</ref>

: <math>f(x_i) = \sum_{fj=1}^n c_j \mathbf K_{ij}, \text{ and } \|f\|^2_{\mathcal H} = \langle f, f\rangle_\mathcal H = \sum_{i=1}^n \sum_{j=1}^n c_i c_jK(x_i, x_j) = c^T \mathbf K c.</math>

==Special properties of the hinge loss==