Online Learning of the Fuzzy Choquet Integral

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 monotonicity and the boundary conditions of the FM have limited its applicability to decision-level fusion. But in a recent work, we removed the boundary and monotonicity constraints of the FM, which we then called a bounded capacity (BC), to propose a Choquet Integral regression (CIR) approach that enables capability beyond previously proposed ChI regression methods. In the same work, we also presented a quadratic programming (QP)-based method, batch-CIR, to learn the BC parameters of the CIR from training data. However, the QP used for learning the BC scales exponentially with the dimensionality of the training data and thus it becomes impractical on data sets with 7 or more dimensions. In this paper we propose an iterative gradient descent approach, online-CIR, to learn the BC. This method iteratively processes the training data, one data point at a time, and therefore requires significantly less computation and space at any time during the training. The application of batch-CIR required the dimensionality reduction of high-dimensional data sets to enable computation in a reasonable time. The proposed online-CIR approach has enabled us to extend CIR to data sets with larger dimensionality. In our experimental evaluation using benchmark regression data sets, online-CIR has outperformed batch-CIR on high-dimensional data sets while also matching the batch-CIR performance on low-dimensional data sets.