The optimum class-selective rejection rule

Class-selective rejection is an extension of simple rejection. That is, when an input pattern cannot be reliably assigned to one of the N classes in an N-class problem, it is assigned to a subset of classes that are most likely to issue the pattern, instead of simply being rejected. By selecting more classes, the risk of making an error can be reduced, at the price of subsequently having a larger remaining number of classes. The optimality of class-selective rejection is therefore defined as the best trade-off between error rate and average number of selected classes. Formally, the trade-off study is embedded in the framework of decision theory. The average expected loss is expressed as a linear combination of error rate and average number of classes. The minimization of the average expected loss, therefore, provides the best trade-off. The complexity of the resulting optimum rule is reduced, via a discrete convex minimization, to be linear in the number of classes. Upper-bounds on error rate and average number of classes are derived. An example is provided to illustrate various aspects of the optimum decision rule. Finally, the implications of the new decision rule are discussed