Regularization perspectives on support vector machines: Difference between revisions

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<math>f = \text{arg}\min_{f\in\mathcal{H}}\left\{\frac{1}{n}\sum_{i=1}^n V(y_i,f(x_i))+\lambda||f||^2_\mathcal{H}\right\} </math>,

where <math>\mathcal{H}</math> is a [[hypothesis space]]<ref>ThisA hypothesis space is the set of functions isused to model the data in a machine learning problem. Each function corresponds to a hypothesis about the structure of the data. Typically the functions in a hypothesis space form a [[Hilbert space]] of allfunctions thewith functionsnorm we'reformed allowingfrom the algorithm toloss pickfunction.</ref> of functions, <math>V:\mathbf Y \times \mathbf Y \to \mathbb R</math> is the loss function, <math>||\cdot||_\mathcal H</math> is a [[norm (mathematics)|norm]] on the hypothesis space of functions, and <math>\lambda\in\mathbb R</math> is the [[regularization parameter]].<ref>For insight on choosing the parameter, see, e.g., {{cite journal|last=Wahba|first=Grace|coauthors=Yonghua Wang|title=When is the optimal regularization parameter insensitive to the choice of the loss function|journal=Communications in Statistics - Theory and Methods|year=1990|volume=19|issue=5|pages=1685-1700|doi=10.1080/03610929008830285|url=http://www.tandfonline.com/doi/abs/10.1080/03610929008830285}}</ref>

When <math>\mathcal{H}</math> is a [[reproducing kernel Hilbert space]], there exists a [[kernel function]] <math>K: \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 journal|last=Scholkopf|first=Bernhard|coauthors=Ralf Herbrich and 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|url=http://www.springerlink.com/content/v1tvba62hd4837h9/?MUD=MP}}</ref> <math>f(x_i) = \sum_{f=1}^n c_j \mathbf K_{ij}</math>, and <math> ||f||^2_{\mathcal H} = \langle f,f\rangle_\mathcal H = \sum_{i=1}^n\sum_{j=1}^n c_ic_jK(x_i,x_j) = c^T\mathbf K c </math>