The data splitting procedure used in the current approach has several potential drawbacks. Firstly, it reduces the effective sample size available for the final independence testing, which can lead to decreased statistical power, particularly in scenarios where the sample size is already limited. This trade-off between training and testing data can hinder the ability to detect subtle dependencies, especially in high-dimensional datasets where more data is crucial for accurate estimation of the HSIC and its variance.
To make the data-splitting procedure more data-efficient, one could explore techniques such as cross-validation, where multiple splits are used to maximize the use of available data while still allowing for robust testing. Additionally, employing bootstrapping methods could help in estimating the distribution of the test statistic without the need for a strict holdout set. Another approach could involve using semi-supervised learning techniques, where the model is trained on both labeled and unlabeled data, thereby leveraging the entire dataset more effectively. Finally, incorporating adaptive sampling strategies that prioritize data points likely to provide the most information about the independence relationship could enhance the efficiency of the learning process.