# Generalized Cauchy-Schwarz Divergence and Its Deep Learning Applications

arXiv: https://arxiv.org/abs/2405.04061 · HTML: https://mlsanity.com/paper/2405.04061/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 65 references that are in the map.
- First posted on arXiv 7 May 2024.

## Builds on (most cited first, 17 of 17)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Moment Matching for Multi-Source Domain Adaptation](https://mlsanity.com/paper/1812.01754/) (2018)
- [Deep Hashing Network for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1706.07522/) (2017)
- [Unsupervised Deep Embedding for Clustering Analysis](https://mlsanity.com/paper/1511.06335/) (2015)
- [Deep Cocktail Network: Multi-source Unsupervised Domain Adaptation with Category Shift](https://mlsanity.com/paper/1803.00830/) (2018)
- [Divergence-aware Federated Self-Supervised Learning](https://mlsanity.com/paper/2204.04385/) (2022)
- [Domain Generalization via Multidomain Discriminant Analysis](https://mlsanity.com/paper/1907.11216/) (2019)
- [Reconsidering Representation Alignment for Multi-view Clustering](https://mlsanity.com/paper/2103.07738/) (2021)
- [Learning to Combine: Knowledge Aggregation for Multi-Source Domain Adaptation](https://mlsanity.com/paper/2007.08801/) (2020)
- [Aligning Domain-specific Distribution and Classifier for Cross-domain Classification from Multiple Sources](https://mlsanity.com/paper/2201.01003/) (2022)
- [Deep Divergence-Based Approach to Clustering](https://mlsanity.com/paper/1902.04981/) (2019)
- [Kernel density estimation based sampling for imbalanced class distribution](https://mlsanity.com/paper/1910.07842/) (2019)
- [Cauchy-Schwarz Regularized Autoencoder](https://mlsanity.com/paper/2101.02149/) (2021)
- [Cauchy-Schwarz Divergence Information Bottleneck for Regression](https://mlsanity.com/paper/2404.17951/) (2024)
- [Recurrent Deep Divergence-based Clustering for simultaneous feature learning and clustering of variable length time series](https://mlsanity.com/paper/1811.12050/) (2018)

## Abstract (arXiv)

Divergence measures play a central role and become increasingly essential in deep learning, yet efficient measures for multiple (more than two) distributions are rarely explored. This becomes particularly crucial in areas where the simultaneous management of multiple distributions is both inevitable and essential. Examples include clustering, multi-source domain adaptation or generalization, and multi-view learning, among others. While computing the mean of pairwise distances between any two distributions is a prevalent method to quantify the total divergence among multiple distributions, it is imperative to acknowledge that this approach is not straightforward and necessitates significant computational resources. In this study, we introduce a new divergence measure tailored for multiple distributions named the generalized Cauchy-Schwarz divergence (GCSD). Additionally, we furnish a kernel-based closed-form sample estimator, making it convenient and straightforward to use in various machine-learning applications. Finally, we explore its profound implications in the realm of deep learning by applying it to tackle two thoughtfully chosen machine-learning tasks: deep clustering and multi-source domain adaptation. Our extensive experimental investigations confirm the robustness and effectiveness of GCSD in both scenarios. The findings also underscore the innovative potential of GCSD and its capability to significantly propel machine learning methodologies that necessitate the quantification of multiple distributions.