Testing conditional independence between two random vectors given a third is a fundamental and challenging problem in statistics, particularly in multivariate nonparametric settings due to the complexity of conditional structures. We propose a novel framework for testing conditional independence using transport maps. At the population level, we show that two well-defined transport maps can transform the conditional independence test into an unconditional independence test, this substantially simplifies the problem. These transport maps are estimated from data using conditional continuous normalizing flow models. Within this framework, we derive a test statistic and prove its consistency under both the null and alternative hypotheses. A permutation-based procedure is employed to evaluate the significance of the test. We validate the proposed method through extensive simulations and real-data analysis. Our numerical studies demonstrate the practical effectiveness of the proposed method for conditional independence testing.

Massive datasets, characterized by both large sample sizes and high-dimensional features, are increasingly prevalent across diverse fields. For example, the 1000 Genomes Project Consortium et al. (2015) study amassed genomic data from 2,504 individuals spanning 26 populations, yielding approximately 12 terabytes data. Often, such datasets are distributed across multiple locations. Fusing data together for centralized statistical analysis is somehow infeasible due to concerns over data privacy, memory and storage limitations, and bandwidth constraints. The absence of fusion centers has thus fueled interest in decentralized distributed learning--a paradigm that fully exploits distributed datasets by performing computations locally. This methodology has found successful applications in fields such as personalized medicine, edge computing, smart utilities, and dimension reduction (Li et al., 2011). A fundamental task in these applications is classification. Penalized support vector machines (SVMs) have been enduringly powerful tools for high-dimensional classification tasks, building on the seminal contributions of Boser et al. (1992) and Vapnik (2000). The standard objective function for penalized SVMs combines the hinge loss with a penalty term.

In the field of crowd counting, the current mainstream CNN-based regression methods simply extract the density information of pedestrians without finding the position of each person. This makes the output of the network often found to contain incorrect responses, which may erroneously estimate the total number and not conducive to the interpretation of the algorithm. To this end, we propose a Bi-Branch Attention Network (BBA-NET) for crowd counting, which has three innovation points. i) A two-branch architecture is used to estimate the density information and location information separately. ii) Attention mechanism is used to facilitate feature extraction, which can reduce false responses. iii) A new density map generation method combining geometric adaptation and Voronoi split is introduced. Our method can integrate the pedestrian's head and body information to enhance the feature expression ability of the density map. Extensive experiments performed on two public datasets show that our method achieves a lower crowd counting error compared to other state-of-the-art methods.