The Set Covering Machine (SCM) was developed with the goal of providing models as generalizable as SVM with sparser models. It does so by selecting the junction (or disjunction) of a subset of Boolean features. The requirement of Boolean features requires a function that converts features to Boolean-valued, if they are not in that form. The main improvement of SCM over previous methods is that it allows to control the trade-off between complexity and accuracy. The VC dimension of the SCM is not defined whenever it uses data-dependent features.

The Set Covering Machine

We won’t usually start from Boolean features, but more likely from an input space $X$ in $\rm I!R^n$. $x$ represents an n-dimensional vector in $X$ containing all the values for a particular feature. We define functions $h_i(x)$ that map $x$ onto ${0,1}$. The goal is, given any set $H = {h_i(x)}_{i=1}^{ H }$ that return a small subset $R \subset H$ of features. Given the subset $R$ and an arbitrary input vector $x$, the output $f(x)$ of the SCM is defined to be

We will say that a feature is consistent with a set of examples if it correctly classifies all the examples in that set.

Valiant defined an algorithm which, given a set of $m$ training examples and a set $H$ of features, found the subset $C \subseteq H$ of all the features which are consistent with all the positive training examples and, in consequence, $\land_{i\in C}h_i(x)$ is consistent with them too. In order for $\land_{i\in C}h_i(x)$ to be consistent with all $m$ training examples, there must exist a subset $E$ of $C$ such that $\land_{i\in E}h_i(x)$ is consistent with all $m$ training examples. However, as $ C $ is likely to be large, we might find subsets of $C$ that return smaller generalization errors.

References

  • Marchand, M., & Shawe-Taylor, J. (2002). The Set Covering Machine. Journal of Machine Learning Research, 3, 723–746. https://doi.org/10.1162/jmlr.2003.3.4-5.723