Editing Support vector machine (section)

=== Transductive support vector machines ===
Transductive support vector machines extend SVMs in that they could also treat partially labeled data in [[semi-supervised learning]] by following the principles of [[Transduction (machine learning)|transduction]]. Here, in addition to the training set <math>\mathcal{D}</math>, the learner is also given a set

<math display="block">\mathcal{D}^\star = \{ \mathbf{x}^\star_i \mid \mathbf{x}^\star_i \in \mathbb{R}^p\}_{i=1}^k </math>

of test examples to be classified. Formally, a transductive support vector machine is defined by the following primal optimization problem:<ref>{{Cite conference |last=Joachims |first=Thorsten |title=Transductive Inference for Text Classification using Support Vector Machines |url=http://www1.cs.columbia.edu/~dplewis/candidacy/joachims99transductive.pdf |conference=Proceedings of the 1999 International Conference on Machine Learning (ICML 1999) |pages=200–209}}</ref>

Minimize (in <math>\mathbf{w}, b, \mathbf{y}^\star</math>)

<math display="block">\frac{1}{2}\|\mathbf{w}\|^2</math>

subject to (for any <math>i = 1, \dots, n</math> and any <math>j = 1, \dots, k</math>)

<math display="block">\begin{align}
&y_i(\mathbf{w} \cdot \mathbf{x}_i - b) \ge 1, \\
&y^\star_j(\mathbf{w} \cdot \mathbf{x}^\star_j - b) \ge 1,
\end{align}</math>

and

<math display="block">y^\star_j \in \{-1, 1\}.</math>

Transductive support vector machines were introduced by Vladimir N. Vapnik in 1998.