# The Conditional Cauchy-Schwarz Divergence with Applications to Time-Series Data and Sequential Decision Making

arXiv: https://arxiv.org/abs/2301.08970 · HTML: https://mlsanity.com/paper/2301.08970/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 100 references that are in the map.
- First posted on arXiv 21 Jan 2023.

## Builds on (most cited first, 29 of 29)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Equality of Opportunity in Supervised Learning](https://mlsanity.com/paper/1610.02413/) (2016)
- [Learning Transferable Features with Deep Adaptation Networks](https://mlsanity.com/paper/1502.02791/) (2015)
- [EEGNet: A Compact Convolutional Network for EEG-based Brain-Computer Interfaces](https://mlsanity.com/paper/1611.08024/) (2016)
- [Deep Transfer Learning with Joint Adaptation Networks](https://mlsanity.com/paper/1605.06636/) (2016)
- [Deep Domain Confusion: Maximizing for Domain Invariance](https://mlsanity.com/paper/1412.3474/) (2014)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Provably Efficient Maximum Entropy Exploration](https://mlsanity.com/paper/1812.02690/) (2018)
- [Learning Robust Representations via Multi-View Information Bottleneck](https://mlsanity.com/paper/2002.07017/) (2020)
- [Mind the Class Weight Bias: Weighted Maximum Mean Discrepancy for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1705.00609/) (2017)
- [A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings](https://mlsanity.com/paper/2002.03689/) (2020)
- [Invariant Information Bottleneck for Domain Generalization](https://mlsanity.com/paper/2106.06333/) (2021)
- [Robust Bi-Tempered Logistic Loss Based on Bregman Divergences](https://mlsanity.com/paper/1906.03361/) (2019)
- [KL Guided Domain Adaptation](https://mlsanity.com/paper/2106.07780/) (2021)
- [A Unifying Framework for Spectrum-Preserving Graph Sparsification and Coarsening](https://mlsanity.com/paper/1902.09702/) (2019)
- [Conditional Generative Moment-Matching Networks](https://mlsanity.com/paper/1606.04218/) (2016)
- [Efficient Conditionally Invariant Representation Learning](https://mlsanity.com/paper/2212.08645/) (2022)
- [FastMMD: Ensemble of Circular Discrepancy for Efficient Two-Sample Test](https://mlsanity.com/paper/1405.2664/) (2014)
- [Discriminative Joint Probability Maximum Mean Discrepancy (DJP-MMD) for Domain Adaptation](https://mlsanity.com/paper/1912.00320/) (2019)
- [Deep Divergence-Based Approach to Clustering](https://mlsanity.com/paper/1902.04981/) (2019)
- [Cauchy-Schwarz Regularized Autoencoder](https://mlsanity.com/paper/2101.02149/) (2021)
- [Direct Estimation of Information Divergence Using Nearest Neighbor Ratios](https://mlsanity.com/paper/1702.05222/) (2017)
- [Time Series Cluster Kernel for Learning Similarities between Multivariate Time Series with Missing Data](https://mlsanity.com/paper/1704.00794/) (2017)
- [Nonsmooth Analysis and Subgradient Methods for Averaging in Dynamic Time Warping Spaces](https://mlsanity.com/paper/1701.06393/) (2017)
- [Self-Supervised Robust Scene Flow Estimation via the Alignment of Probability Density Functions](https://mlsanity.com/paper/2203.12193/) (2022)
- [Direct coupling information measure from non-uniform embedding](https://mlsanity.com/paper/1305.6204/) (2013)
- [Guided Policy Exploration for Markov Decision Processes using an Uncertainty-Based Value-of-Information Criterion](https://mlsanity.com/paper/1802.01518/) (2018)
- [Estimating Kullback-Leibler Divergence Using Kernel Machines](https://mlsanity.com/paper/1905.00586/) (2019)
- [From Time Series to Networks in R with the ts2net Package](https://mlsanity.com/paper/2208.09660/) (2022)

## Built on by (most cited first, 6 of 6)

- [Cauchy-Schwarz Divergence Information Bottleneck for Regression](https://mlsanity.com/paper/2404.17951/) (2024)
- [Distributional Vision-Language Alignment by Cauchy-Schwarz Divergence](https://mlsanity.com/paper/2502.17028/) (2025)
- [Domain Adaptation with Cauchy-Schwarz Divergence](https://mlsanity.com/paper/2405.19978/) (2024)
- [Explainable Multimodal Regression via Information Decomposition](https://mlsanity.com/paper/2512.22102/) (2025)
- [Brain foundation model-guided source-selective domain adaptation for cross-subject EEG decoding](https://mlsanity.com/paper/2507.21037/) (2025)
- [Cross-Modal Retrieval with Cauchy-Schwarz Divergence](https://mlsanity.com/paper/2509.21339/) (2025)

## Abstract (arXiv)

The Cauchy-Schwarz (CS) divergence was developed by Pr'incipe et al. in 2000. In this paper, we extend the classic CS divergence to quantify the closeness between two conditional distributions and show that the developed conditional CS divergence can be simply estimated by a kernel density estimator from given samples. We illustrate the advantages (e.g., rigorous faithfulness guarantee, lower computational complexity, higher statistical power, and much more flexibility in a wide range of applications) of our conditional CS divergence over previous proposals, such as the conditional KL divergence and the conditional maximum mean discrepancy. We also demonstrate the compelling performance of conditional CS divergence in two machine learning tasks related to time series data and sequential inference, namely time series clustering and uncertainty-guided exploration for sequential decision making. The code of conditional CS divergence is available at https://github.com/SJYuCNEL/conditionalCSdivergence.