Multiple Instance Mamdani Fuzzy Inference

Abstract

A novel fuzzy learning framework that employs fuzzy inference to solve the problem of Multiple Instance Learning (MIL) is presented. The framework introduces a new class of fuzzy inference systems called Multiple Instance Mamdani Fuzzy Inference Systems (MIMamdani). In multiple instance problems, the training data is ambiguously labeled. Instances are grouped into bags, labels of bags are known but not those of individual instances. MIL deals with learning a classifier at the bag level. Over the years, many solutions to this problem have been proposed. However, no MIL formulation employing fuzzy inference exists in the literature. Fuzzy logic is powerful at modeling knowledge uncertainty and measurements imprecision. It is one of the best frameworks to model vagueness. However, in addition to uncertainty and imprecision, there is a third vagueness concept that fuzzy logic does not address quiet well, yet. This vagueness concept is due to the ambiguity that arises when the data have multiple forms of expression, this is the case for multiple instance problems. In this paper, we introduce multiple instance fuzzy logic that enables fuzzy reasoning with bags of instances. Accordingly, a MI-Mamdani that extends the standard Mamdani inference system to compute with multiple instances is introduced. The proposed framework is tested and validated using a synthetic dataset suitable for MIL problems. Additionally, we apply the proposed multiple instance inference to fuse the output of multiple discrimination algorithms for the purpose of landmine detection using Ground Penetrating Radar.