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 2N variables for N inputs and N(2N-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.

中文翻译：

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Choquet 积分的数据驱动压缩和高效学习


Choquet 积分 (ChI) 是根据模糊测度 (FM) 定义的参数非线性聚合函数。遗憾的是，迄今为止，ChI 的应用仅限于解决输入数量相对较少的问题。主要是因为 FM 有 N 个输入的 2N 个变量和 N(2N-1 - 1) 个单调性约束。作为回报，社区已转向基于密度的插补（例如，Sugeno λ-FM）或相互作用（FM 变量）的数量受到限制（例如，k-可加性）。在这里，我们提出了一种新的可扩展的数据驱动方法来表示和学习 ChI，使得学习可以在计算上管理更大的 N。首先，识别数据支持的变量并在优化中使用。识别这些变量还使我们能够识别未来不适定的融合场景；涉及数据不支持的变量子集的 ChI。其次，我们概述了一个插补函数框架来解决数据不支持的变量。第三，我们提出了一种无损方法来压缩冗余变量和相关的单调性约束。最后，我们概述了一种有损近似方法来进一步压缩 ChI（如果/需要时）。对具有已知 FM 的合成数据集进行的计算复杂性分析和实验证明了所提出理论的有效性和效率。