Regularized Optimal Transport Layers for Generalized Global Pooling Operations

—Global pooling is one of the most signiﬁcant operations in many machine learning models and tasks, which works for information fusion and structured data (like sets and graphs) representation. However, without solid mathematical fundamentals, its practical implementations often depend on empirical mechanisms and thus lead to sub-optimal, even unsatisfactory performance. In this work, we develop a novel and generalized global pooling framework through the lens of optimal transport. The proposed framework is interpretable from the perspective of expectation-maximization. Essentially, it aims at learning an optimal transport across sample indices and feature dimensions, making the corresponding pooling operation maximize the conditional expectation of input data. We demonstrate that most existing pooling methods are equivalent to solving a regularized optimal transport (ROT) problem with different specializations, and more sophisticated pooling operations can be implemented by hierarchically solving multiple ROT problems. Making the parameters of the ROT problem learnable, we develop a family of regularized optimal transport pooling (ROTP) layers. We implement the ROTP layers as a new kind of deep implicit layer. Their model architectures correspond to different optimization algorithms. We test our ROTP layers in several representative set-level machine learning scenarios, including multi-instance learning (MIL), graph classiﬁcation, graph set representation, and image classiﬁcation. Experimental results show that applying our ROTP layers can reduce the difﬁculty of the design and selection of global pooling — our ROTP layers may either imitate some existing global pooling methods or lead to some new pooling layers ﬁtting data better. The code is available at https://github.com/SDS-Lab/ROT-Pooling.