Group fairness in machine learning is often enforced by adding a regularizer that reduces the dependence between model predictions and sensitive attributes. However, existing regularizers are built on heterogeneous distance measures and design choices, which makes their behavior hard to reason about and their performance inconsistent across tasks. This raises a basic question: what properties make a good fairness regularizer? We address this question by first organizing existing in-process methods into three families: (i) matching prediction statistics across sensitive groups, (ii) aligning latent representations, and (iii) directly minimizing dependence between predictions and sensitive attributes. Through this lens, we identify desirable properties of the underlying distance measure, including tight generalization bounds, robustness to scale differences, and the ability to handle arbitrary prediction distributions. Motivated by these properties, we propose a Cauchy-Schwarz (CS) fairness regularizer that penalizes the empirical CS divergence between prediction distributions conditioned on sensitive groups. Under a Gaussian comparison, we show that CS divergence yields a tighter bound than Kullback-Leibler divergence, Maximum Mean Discrepancy, and the mean disparity used in Demographic Parity, and we discuss how these advantages translate to a distribution-free, kernel-based estimator that naturally extends to multiple sensitive attributes. Extensive experiments on four tabular benchmarks and one image dataset demonstrate that the proposed CS regularizer consistently improves Demographic Parity and Equal Opportunity metrics while maintaining competitive accuracy, and achieves a more stable utility-fairness trade-off across hyperparameter settings compared to prior regularizers.

Effective cross-modal retrieval requires robust alignment of heterogeneous data types. Most existing methods focus on bi-modal retrieval tasks and rely on distributional alignment techniques such as Kullback-Leibler divergence, Maximum Mean Discrepancy, and correlation alignment. However, these methods often suffer from critical limitations, including numerical instability, sensitivity to hyperparameters, and their inability to capture the full structure of the underlying distributions. In this paper, we introduce the Cauchy-Schwarz (CS) divergence, a hyperparameter-free measure that improves both training stability and retrieval performance. We further propose a novel Generalized CS (GCS) divergence inspired by Hölder's inequality. This extension enables direct alignment of three or more modalities within a unified mathematical framework through a bidirectional circular comparison scheme, eliminating the need for exhaustive pairwise comparisons. Extensive experiments on six benchmark datasets demonstrate the effectiveness of our method in both bi-modal and tri-modal retrieval tasks. The code of our CS/GCS divergence is publicly available at https://github.com/JiahaoZhang666/CSD.

--Decoding motor imagery (MI) electroencephalogram (EEG) signals, a key non-invasive brain-computer interface (BCI) paradigm for controlling external systems, has been significantly advanced by deep learning. However, MI-EEG decoding remains challenging due to substantial inter-subject variability and limited labeled target data, which necessitate costly calibration for new users. Many existing multi-source domain adaptation (MSDA) methods indiscriminately incorporate all available source domains, disregarding the large inter-subject differences in EEG signals, which leads to negative transfer and excessive computational costs. Moreover, while many approaches focus on feature distribution alignment, they often neglect the explicit dependence between features and decision-level outputs, limiting their ability to preserve discriminative structures. T o address these gaps, we propose a novel MSDA framework that leverages a pretrained large Brain Foundation Model (BFM) for dynamic and informed source subject selection, ensuring only relevant sources contribute to adaptation. Furthermore, we employ Cauchy-Schwarz (CS) and Conditional CS (CCS) divergences to jointly perform feature-level and decision-level alignment, enhancing domain invariance while maintaining class discriminability. Extensive evaluations on two benchmark MI-EEG datasets demonstrate that our framework outperforms a broad range of state-of-the-art baselines. Additional experiments with a large source pool validate the scalability and efficiency of BFM-guided selection, which significantly reduces training time without sacrificing performance. RAIN-COMPUTER interfaces (BCIs) establish a direct communication channel between the brain and external systems by interpreting neural activity without relying on neuromuscular pathways [1].

Multimodal alignment is crucial for various downstream tasks such as cross-modal generation and retrieval. Previous multimodal approaches like CLIP maximize the mutual information mainly by aligning pairwise samples across modalities while overlooking the distributional differences, leading to suboptimal alignment with modality gaps. In this paper, to overcome the limitation, we propose CS-Aligner, a novel and straightforward framework that performs distributional vision-language alignment by integrating Cauchy-Schwarz (CS) divergence with mutual information. In the proposed framework, we find that the CS divergence and mutual information serve complementary roles in multimodal alignment, capturing both the global distribution information of each modality and the pairwise semantic relationships, yielding tighter and more precise alignment. Moreover, CS-Aligher enables incorporating additional information from unpaired data and token-level representations, enhancing flexible and fine-grained alignment in practice. Experiments on text-to-image generation and cross-modality retrieval tasks demonstrate the effectiveness of our method on vision-language alignment.

Domain adaptation aims to use training data from one or multiple source domains to learn a hypothesis that can be generalized to a different, but related, target domain. As such, having a reliable measure for evaluating the discrepancy of both marginal and conditional distributions is crucial. We introduce Cauchy-Schwarz (CS) divergence to the problem of unsupervised domain adaptation (UDA). The CS divergence offers a theoretically tighter generalization error bound than the popular Kullback-Leibler divergence. This holds for the general case of supervised learning, including multi-class classification and regression. Furthermore, we illustrate that the CS divergence enables a simple estimator on the discrepancy of both marginal and conditional distributions between source and target domains in the representation space, without requiring any distributional assumptions. We provide multiple examples to illustrate how the CS divergence can be conveniently used in both distance metric- or adversarial training-based UDA frameworks, resulting in compelling performance.

The information bottleneck (IB) approach is popular to improve the generalization, robustness and explainability of deep neural networks. Essentially, it aims to find a minimum sufficient representation t by striking a trade-off between a compression term I(x; t) and a prediction term I(y; t), where I(;) refers to the mutual information (MI). MI is for the IB for the most part expressed in terms of the Kullback-Leibler (KL) divergence, which in the regression case corresponds to prediction based on mean squared error (MSE) loss with Gaussian assumption and compression approximated by variational inference. In this paper, we study the IB principle for the regression problem and develop a new way to parameterize the IB with deep neural networks by exploiting favorable properties of the Cauchy-Schwarz (CS) divergence. By doing so, we move away from MSE-based regression and ease estimation by avoiding variational approximations or distributional assumptions. We investigate the improved generalization ability of our proposed CS-IB and demonstrate strong adversarial robustness guarantees. We demonstrate its superior performance on six real-world regression tasks over other popular deep IB approaches. We additionally observe that the solutions discovered by CS-IB always achieve the best trade-off between prediction accuracy and compression ratio in the information plane. The code is available at https://github.com The information bottleneck (IB) principle was proposed by (Tishby et al., 1999) as an informationtheoretic framework for representation learning. It considers extracting information about a target variable y through a correlated variable x. The extracted information is characterized by another variable t, which is (a possibly randomized) function of x. Formally, the IB objective is to learn a representation t that maximizes its predictive power to y subject to some constraints on the amount of information that it carries about x: max I(y; t) s.t.

The Cauchy-Schwarz (CS) divergence was developed by Pr\'{i}ncipe 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., the rigorous faithfulness guarantee, the lower computational complexity, the higher statistical power, and the 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 the time series clustering and the uncertainty-guided exploration for sequential decision making.