Hybrid Gromov–Wasserstein Embedding for Capsule Learning

Capsule networks (CapsNets) aim to parse images into a hierarchy of objects, parts, and their relationships using a two-step process involving part–whole transformation and hierarchical component routing. However, this hierarchical relationship modeling is computationally expensive, which has limited the wider use of CapsNet despite its potential advantages. The current state of CapsNet models primarily focuses on comparing their performance with capsule baselines, falling short of achieving the same level of proficiency as deep convolutional neural network (CNN) variants in intricate tasks. To address this limitation, we present an efficient approach for learning capsules that surpasses canonical baseline models and even demonstrates superior performance compared with high-performing convolution models. Our contribution can be outlined in two aspects: first, we introduce a group of subcapsules onto which an input vector is projected. Subsequently, we present the hybrid Gromov–Wasserstein (HGW) framework, which initially quantifies the dissimilarity between the input and the components modeled by the subcapsules, followed by determining their alignment degree through optimal transport (OT). This innovative mechanism capitalizes on new insights into defining alignment between the input and subcapsules, based on the similarity of their respective component distributions. This approach enhances CapsNets’ capacity to learn from intricate, high-dimensional data while retaining their interpretability and hierarchical structure. Our proposed model offers two distinct advantages: 1) its lightweight nature facilitates the application of capsules to more intricate vision tasks, including object detection; and 2) it outperforms baseline approaches in these demanding tasks. Our empirical findings illustrate that HGW capsules (HGWCapsules) exhibit enhanced robustness against affine transformations, scale effectively to larger datasets, and surpass CNN and CapsNet models across various vision tasks.