Online Sequential Learning of Fuzzy Measures for Choquet Integral Fusion

The Choquet integral (ChI) is an aggregation operator defined with respect to a fuzzy measure (FM). The FM encodes the worth of all subsets of the sources of information that are being aggregated. The ChI is capable of representing many aggregation functions and has found its application in a wide range of decision fusion problems. In our prior work, we introduced a data support-based approach for learning the FM for decision fusion problems. This approach applies a quadratic programming (QP)-based method to train the FM. However, since the FM of ChI scales as $2^{N}$, where $N$ is the number of input sources, the space complexity for learning the FM grows exponentially with $N$. This has limited the practical application of ChI-based decision fusion methods to small numbers of dimenstions—$N$ ≲ 6 is practical in most cases. In this work, we propose an iterative gradient descent-based approach to train the FM for ChI with an efficient method for handling the FM constraints. This method processes the training data, one observation at a time, and thereby significantly reduces the space complexity of the training process. We tested our online method on synthetic and real-world data sets, and compared the performance and convergence behaviour with our previously proposed QP-based method (i.e., batch method). On 10 out of 12 data sets, the online learning method has either matched or outperformed the batch method. We also show that we are able to use larger numbers of inputs with the online learning approach, extending the practical application of the ChI.