# Hybrid Gromov-Wasserstein Embedding for Capsule Learning

arXiv: https://arxiv.org/abs/2209.00232 · HTML: https://mlsanity.com/paper/2209.00232/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 90 references that are in the map.
- First posted on arXiv 1 Sep 2022.

## Builds on (most cited first, 30 of 46)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Mask R-CNN](https://mlsanity.com/paper/1703.06870/) (2017)
- [Squeeze-and-Excitation Networks](https://mlsanity.com/paper/1709.01507/) (2017)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Rethinking Atrous Convolution for Semantic Image Segmentation](https://mlsanity.com/paper/1706.05587/) (2017)
- [Microsoft COCO Captions: Data Collection and Evaluation Server](https://mlsanity.com/paper/1504.00325/) (2015)
- [SuperPoint: Self-Supervised Interest Point Detection and Description](https://mlsanity.com/paper/1712.07629/) (2017)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [Skin Lesion Analysis Toward Melanoma Detection 2018: A Challenge Hosted by the International Skin Imaging Collaboration (ISIC)](https://mlsanity.com/paper/1902.03368/) (2019)
- [A Survey of Convolutional Neural Networks: Analysis, Applications, and Prospects](https://mlsanity.com/paper/2004.02806/) (2020)
- [Attention Augmented Convolutional Networks](https://mlsanity.com/paper/1904.09925/) (2019)
- [DISK: Learning local features with policy gradient](https://mlsanity.com/paper/2006.13566/) (2020)
- [Gromov-Wasserstein Learning for Graph Matching and Node Embedding](https://mlsanity.com/paper/1901.06003/) (2019)
- [How to represent part-whole hierarchies in a neural network](https://mlsanity.com/paper/2102.12627/) (2021)
- [The One Hundred Layers Tiramisu: Fully Convolutional DenseNets for Semantic Segmentation](https://mlsanity.com/paper/1611.09326/) (2016)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [Scalable Gromov-Wasserstein Learning for Graph Partitioning and Matching](https://mlsanity.com/paper/1905.07645/) (2019)
- [Stacked Capsule Autoencoders](https://mlsanity.com/paper/1906.06818/) (2019)
- [Wasserstein Embedding for Graph Learning](https://mlsanity.com/paper/2006.09430/) (2020)
- [The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation](https://mlsanity.com/paper/2009.04266/) (2020)
- [Learning Generative Models across Incomparable Spaces](https://mlsanity.com/paper/1905.05461/) (2019)
- [GIFT: Learning Transformation-Invariant Dense Visual Descriptors via Group CNNs](https://mlsanity.com/paper/1911.05932/) (2019)
- [CO-Optimal Transport](https://mlsanity.com/paper/2002.03731/) (2020)
- [Efficient-CapsNet: Capsule Network with Self-Attention Routing](https://mlsanity.com/paper/2101.12491/) (2021)
- [Capsules for Object Segmentation](https://mlsanity.com/paper/1804.04241/) (2018)
- [Capsules with Inverted Dot-Product Attention Routing](https://mlsanity.com/paper/2002.04764/) (2020)

## Built on by (most cited first, 1 of 1)

- [Fractional Correspondence Framework in Detection Transformer](https://mlsanity.com/paper/2503.04107/) (2025)

## Abstract (arXiv)

Capsule networks (CapsNets) aim to parse images into a hierarchy of objects, parts, and their relations 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 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 to high-performing convolution models. Our contribution can be outlined in two aspects: firstly, we introduce a group of subcapsules onto which an input vector is projected. Subsequently, we present the Hybrid Gromov-Wasserstein 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. 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: (i) its lightweight nature facilitates the application of capsules to more intricate vision tasks, including object detection; (ii) it outperforms baseline approaches in these demanding tasks.