Data-Driven Compression and Efficient Learning of the Choquet Integral

The Choquet integral (ChI) is a parametric nonlinear aggregation function defined with respect to the fuzzy measure (FM). To date, application of the ChI has sadly been restricted to problems with relatively few numbers of inputs; primarily as the FM has 2^N variables for N inputs and N(2^N-1 - 1) monotonicity constraints. In return, the community has turned to density-based imputation (e.g., Sugeno λ-FM) or the number of interactions (FM variables) are restricted (e.g., k-additivity). Herein, we propose a new scalable data-driven way to represent and learn the ChI, making learning computationally manageable for larger N. First, data supported variables are identified and used in optimization. Identification of these variables also allows us recognize future ill-posed fusion scenarios; ChIs involving variable subsets not supported by data. Second, we outline an imputation function framework to address data unsupported variables. Third, we present a lossless way to compress redundant variables and associated monotonicity constraints. Finally, we outline a lossy approximation method to further compress the ChI (if/when desired). Computational complexity analysis and experiments conducted on synthetic datasets with known FMs demonstrate the effectiveness and efficiency of the proposed theory.