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 <inline-formula><tex-math notation="LaTeX">$2^N$</tex-math></inline-formula> variables for <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> inputs and <inline-formula> <tex-math notation="LaTeX">$N(2^{N-1}-1)$</tex-math></inline-formula> monotonicity constraints. In return, the community has turned to density-based imputation (e.g., Sugeno <inline-formula><tex-math notation="LaTeX">$\lambda$ </tex-math></inline-formula>-FM) or the number of <italic>interactions</italic> (FM variables) are restricted (e.g., <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-additivity). Herein, we propose a new scalable data-driven way to represent and learn the ChI, making learning computationally manageable for larger <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula>. 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.