Graph Neural Networks (GNNs) have shown significant success for graph-based tasks. Motivated by the prevalence of large datasets in real-world applications, pooling layers are crucial components of GNNs. By reducing the size of input graphs, pooling enables faster training and potentially better generalisation.

The rapid growth of high-dimensional datasets across various scientific domains has created a pressing need for new statistical methods to compare distributions supported on their underlying structures. Assessing similarity between datasets whose samples lie on low-dimensional manifolds requires robust techniques capable of separating meaningful signal from noise. We propose a principled framework for statistical inference of similarity and alignment between distributions supported on manifolds underlying high-dimensional datasets in the presence of heterogeneous noise. The key idea is to link the low-rank structure of observed data matrices to their underlying manifold geometry. By analyzing the spectrum of the sample covariance under a manifold signal-plus-noise model, we develop a scale-invariant distance measure between datasets based on their principal variance structures. We further introduce a consistent estimator for this distance and a statistical test for manifold alignability, and establish their asymptotic properties using random matrix theory. The proposed framework accommodates heterogeneous noise across datasets and offers an efficient, theoretically grounded approach for comparing high-dimensional datasets with low-dimensional manifold structures. Through extensive simulations and analyses of multi-sample single-cell datasets, we demonstrate that our method achieves superior robustness and statistical power compared with existing approaches.

Machine Learning Force Fields (MLFFs) are a promising alternative to expensive ab initio quantum mechanical molecular simulations. Given the diversity of chemical spaces that are of interest and the cost of generating new data, it is important to understand how MLFFs generalize beyond their training distributions. In order to characterize and better understand distribution shifts in MLFFs, we conduct diagnostic experiments on chemical datasets, revealing common shifts that pose significant challenges, even for large foundation models trained on extensive data. Based on these observations, we hypothesize that current supervised training methods inadequately regularize MLFFs, resulting in overfitting and learning poor representations of out-of-distribution systems. We then propose two new methods as initial steps for mitigating distribution shifts for MLFFs. Our methods focus on test-time refinement strategies that incur minimal computational cost and do not use expensive ab initio reference labels. The first strategy, based on spectral graph theory, modifies the edges of test graphs to align with graph structures seen during training. Our second strategy improves representations for out-of-distribution systems at test-time by taking gradient steps using an auxiliary objective, such as a cheap physical prior. Our test-time refinement strategies significantly reduce errors on out-of-distribution systems, suggesting that MLFFs are capable of and can move towards modeling diverse chemical spaces, but are not being effectively trained to do so. Our experiments establish clear benchmarks for evaluating the generalization capabilities of the next generation of MLFFs. Our code is available at https://tkreiman.github.io/projects/mlff_distribution_shifts/.

Heterogeneous graphs can well describe the complex entity relationships in the real world. For example, online shopping networks contain multiple physical types of consumers and products, as well as multiple relationship types such as purchasing and favoriting. More and more scholars pay attention to this research because heterogeneous graph representation learning shows strong application potential in real-world scenarios. However, the existing heterogeneous graph models use data augmentation techniques to enhance the use of graph structure information, which only captures the graph structure information from the spatial topology, ignoring the information displayed in the spectrum dimension of the graph structure. To address the issue that heterogeneous graph representation learning methods fail to model spectral information, this paper introduces a spectral-enhanced graph contrastive learning model (SHCL) and proposes a spectral augmentation algorithm for the first time in heterogeneous graph neural networks. The proposed model learns an adaptive topology augmentation scheme through the heterogeneous graph itself, disrupting the structural information of the heterogeneous graph in the spectrum dimension, and ultimately improving the learning effect of the model. Experimental results on multiple real-world datasets demonstrate substantial advantages of the proposed model.

Increasing the throughput of the Transformer architecture, a foundational component used in numerous state-of-the-art models for vision and language tasks (e.g., GPT, LLaVa), is an important problem in machine learning. One recent and effective strategy is to merge token representations within Transformer models, aiming to reduce computational and memory requirements while maintaining accuracy. Prior works have proposed algorithms based on Bipartite Soft Matching (BSM), which divides tokens into distinct sets and merges the top k similar tokens. However, these methods have significant drawbacks, such as sensitivity to token-splitting strategies and damage to informative tokens in later layers. This paper presents a novel paradigm called PiToMe, which prioritizes the preservation of informative tokens using an additional metric termed the energy score. This score identifies large clusters of similar tokens as high-energy, indicating potential candidates for merging, while smaller (unique and isolated) clusters are considered as low-energy and preserved. Experimental findings demonstrate that PiToMe saved from 40-60\% FLOPs of the base models while exhibiting superior off-the-shelf performance on image classification (0.5\% average performance drop of ViT-MAE-H compared to 2.6\% as baselines), image-text retrieval (0.3\% average performance drop of CLIP on Flickr30k compared to 4.5\% as others), and analogously in visual questions answering with LLaVa-7B. Furthermore, PiToMe is theoretically shown to preserve intrinsic spectral properties of the original token space under mild conditions

Time-series analysis plays a pivotal role across a range of critical applications, from finance to healthcare, which involves various tasks, such as forecasting and classification. To handle the inherent complexities of time-series data, such as high dimensionality and noise, traditional supervised learning methods first annotate extensive labels for time-series data in each task, which is very costly and impractical in real-world applications. In contrast, pre-trained foundation models offer a promising alternative by leveraging unlabeled data to capture general time series patterns, which can then be fine-tuned for specific tasks. However, existing approaches to pre-training such models typically suffer from high-bias and low-generality issues due to the use of predefined and rigid augmentation operations and domain-specific data training. To overcome these limitations, this paper introduces UniCL, a universal and scalable contrastive learning framework designed for pretraining time-series foundation models across cross-domain datasets. Specifically, we propose a unified and trainable time-series augmentation operation to generate pattern-preserved, diverse, and low-bias time-series data by leveraging spectral information. Besides, we introduce a scalable augmentation algorithm capable of handling datasets with varying lengths, facilitating cross-domain pretraining. Extensive experiments on two benchmark datasets across eleven domains validate the effectiveness of UniCL, demonstrating its high generalization on time-series analysis across various fields.