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.

The integration of Time-of-Flight (TOF) information in the reconstruction process of Positron Emission Tomography (PET) yields improved image properties. However, implementing the cutting-edge model-based deep learning methods for TOF-PET reconstruction is challenging due to the substantial memory requirements. In this study, we present a novel model-based deep learning approach, LMPDNet, for TOF-PET reconstruction from list-mode data. We address the issue of real-time parallel computation of the projection matrix for list-mode data, and propose an iterative model-based module that utilizes a dedicated network model for list-mode data. Our experimental results indicate that the proposed LMPDNet outperforms traditional iteration-based TOF-PET list-mode reconstruction algorithms. Additionally, we compare the spatial and temporal consumption of list-mode data and sinogram data in model-based deep learning methods, demonstrating the superiority of list-mode data in model-based TOF-PET reconstruction.