This article proposes a new method for causal discovery based on a novel dependence measurement criterion, namely, Multi-level Wavelet Mapping Correlation (MWMC). MWMC captures nonlinear dependencies between variables by measuring their correlations across multiple levels of wavelet mappings. From a theoretical perspective, we show that the empirical estimate of MWMC converges exponentially fast to its population quantity. Under the null hypothesis of independence, we further design a permutation-based independence testing procedure, termed the Wavelet Independence Test (WIT), built upon MWMC. We prove that WIT not only effectively controls the Type I error rate (false positives), but also guarantees that the Type II error rate (false negatives) is upper bounded by \(\mathcal{O}(n^{-1})\) , where \( n \) denotes the sample size, even with a finite number of permutations. Building on these theoretical guarantees, we derive a causal discovery method by integrating MWMC-based WIT into standard causal discovery pipelines. Extensive experiments on (conditional) independence testing and causal discovery using both synthetic and real-world datasets with varying sample sizes demonstrate that our approach consistently outperforms existing independence testing and causal discovery methods in terms of reduced Type II error rates and statistically validated performance improvements. Impact Statement —Causal discovery is a fundamental task in knowledge discovery, aiming to uncover the underlying data-generating mechanisms in order to support more accurate and interpretable predictions. Statistical independence tests and conditional independence (CI) tests have long served as core tools in this area. To improve the reliability of independence testing, we propose a novel test, WIT, which achieves lower Type II error rates in 19 out of 25 distinct experimental scenarios involving diverse data distributions, compared to 15 out of 25 for the strongest existing baseline. We further apply WIT to CI testing and causal discovery, and extensive empirical results show that it consistently improves the performance of multiple causal discovery algorithms across a range of experimental settings.

Determining conditional independence (CI) relationships between random variables is a fundamental yet challenging task in machine learning and statistics, especially in high-dimensional settings. Existing generative model-based CI testing methods, such as those utilizing generative adversarial networks (GANs), often struggle with undesirable modeling of conditional distributions and training instability, resulting in subpar performance. To address these issues, we propose a novel CI testing method via score-based generative modeling, which achieves precise Type I error control and strong testing power. Concretely, we first employ a sliced conditional score matching scheme to accurately estimate conditional score and use Langevin dynamics conditional sampling to generate null hypothesis samples, ensuring precise Type I error control. Then, we incorporate a goodness-of-fit stage into the method to verify generated samples and enhance interpretability in practice. We theoretically establish the error bound of conditional distributions modeled by score-based generative models and prove the validity of our CI tests. Extensive experiments on both synthetic and real-world datasets show that our method significantly outperforms existing state-of-the-art methods, providing a promising way to revitalize generative model-based CI testing.

Graph neural networks (GNNs) are prone to catastrophic forgetting of past experience in continuous learning scenarios. In this work, we propose a novel method for class-incremental graph learning (CGL) by class prototype construction and augmentation, which can effectively overcome catastrophic forgetting and requires no storage of exemplars (i.e., data-free). Concretely, on the one hand, we construct class prototypes in the embedding space that contain rich topological information of nodes or graphs to represent past data, which are then used for future learning. On the other hand, to boost the adaptability of the model to new classes, we employ class prototype augmentation (PA) to create virtual classes by combining current prototypes. Theoretically, we show that PA can promote the model's adaptation to new data and reduce the inconsistency of old prototypes in the embedding space, therefore further mitigate catastrophic forgetting. Extensive experiments on both node and graph classification datasets show that our method significantly outperforms the existing methods in reducing catastrophic forgetting, and beats the existing methods in most cases in terms of classification accuracy.

Discovering causal relationships from observational data is a challenging topic in artificial intelligence. Recent works formulate causal discovery as a continuous optimization problem with a differentiable acyclic constraint. Although these methods have achieved considerable performance improvement, they have two drawbacks: 1) they require a relatively large number of training samples; and 2) their performance will substantially deteriorate when facing heterogeneous noise. To address these problems, we first propose a low-order conditional independence (CI) constraint for the continuous optimization problem, and then design a soft version of the constraint by transforming it to a regularization term in the loss function of the continuous optimization problem. We show the convergence of continuous optimization with our constraint under some mild conditions, and the consistency of causal structure learning with the CI regularization. Extensive experiments on both synthetic and real-world datasets show that with our CI constraint or regularization, existing continuous optimization methods can achieve considerable performance improvement of causal discovery, especially when sample size is small.