Efficient binary fuzzy measure representation and Choquet integral learning - Islam, Muhammad Aminul; Anderson, Derek T.; Du, Xiaoxiao; Havens, Timothy C.; Wagner, Christian (2018)
The Choquet integral (ChI), a parametric function for information aggregation, is parameterized by the fuzzy measure (FM), which has 2N real-valued variables for N inputs. However, the ChI incurs huge storage and computational burden due to its exponential complexity relative to N and, as a result, its calculation, storage, and learning becomes intractable for even modest sizes (e.g., N = 15). Inspired by empirical observations in multi-sensor fusion and the more general need to mitigate the storage, computational, and learning limitations, we previously explored the binary ChI (BChI) relative to the binary fuzzy measure (BFM). The BChI is a natural _t for many applications and can be used to approximate others. Previously, we investigated different properties of the BChI and we provided an initial representation. In this article, we propose a new efficient learning algorithm for the BChI, called EBChI, by utilizing the BFM properties that add at most one variable per training instance. Furthermore, we provide an efficient representation of the BFM (EBFM) scheme that further reduces the number of variables required for storage and computation, thus enabling the use of the BChI for \big N". Finally, we conduct experiments on synthetic data that demonstrate the efficiency of our proposed techniques.