Graph neural networks (GNNs) -- learn graph representations by exploiting the graph's sparsity, connectivity, and symmetries -- have become indispensable for learning geometric data like molecules. However, the most used graphs (e.g., radial cutoff graphs) in molecular modeling lack theoretical guarantees for achieving connectivity and sparsity simultaneously, which are essential for the performance and scalability of GNNs. Furthermore, existing widely used graph construction methods for molecules lack rigidity, limiting GNNs' ability to exploit graph nodes' spatial arrangement. In this paper, we introduce a new hyperparameter-free graph construction of molecules and beyond with sparsity, connectivity, and rigidity guarantees. Remarkably, our method consistently generates connected and sparse graphs with the edge-to-node ratio being bounded above by 3. Our graphs' rigidity guarantees that edge distances and dihedral angles are sufficient to uniquely determine the general spatial arrangements of atoms. We substantiate the effectiveness and efficiency of our proposed graphs in various molecular modeling benchmarks. Code is available at https://github.com/shihhsinwang0214/SCHull.

Conditional flow matching (CFM) stands out as an efficient, simulation-free approach for training flow-based generative models, achieving remarkable performance for data generation. However, CFM is insufficient to ensure accuracy in learning probability paths. In this paper, we introduce a new partial differential equation characterization for the error between the learned and exact probability paths, along with its solution. We show that the total variation gap between the two probability paths is bounded above by a combination of the CFM loss and an associated divergence loss. This theoretical insight leads to the design of a new objective function that simultaneously matches the flow and its divergence. Our new approach improves the performance of the flow-based generative model by a noticeable margin without sacrificing generation efficiency. We showcase the advantages of this enhanced training approach over CFM on several important benchmark tasks, including generative modeling for dynamical systems, DNA sequences, and videos. Code is available at https://github.com/Utah-Math-Data-Science/Flow_Div_Matching.

Graph neural networks (GNNs) have emerged as powerful tools for learning protein structures by capturing spatial relationships at the residue level. However, existing GNN-based methods often face challenges in learning multiscale representations and modeling long-range dependencies efficiently. In this work, we propose an efficient multiscale graph-based learning framework tailored to proteins. Our proposed framework contains two crucial components: (1) It constructs a hierarchical graph representation comprising a collection of fine-grained subgraphs, each corresponding to a secondary structure motif (e.g., $\alpha$-helices, $\beta$-strands, loops), and a single coarse-grained graph that connects these motifs based on their spatial arrangement and relative orientation. (2) It employs two GNNs for feature learning: the first operates within individual secondary motifs to capture local interactions, and the second models higher-level structural relationships across motifs. Our modular framework allows a flexible choice of GNN in each stage. Theoretically, we show that our hierarchical framework preserves the desired maximal expressiveness, ensuring no loss of critical structural information. Empirically, we demonstrate that integrating baseline GNNs into our multiscale framework remarkably improves prediction accuracy and reduces computational cost across various benchmarks.

Theoretical and empirical comparisons have been made to assess the expressive power and performance of invariant and equivariant GNNs. However, there is currently no theoretical result comparing the expressive power of $k$-hop invariant GNNs and equivariant GNNs. Additionally, little is understood about whether the performance of equivariant GNNs, employing steerable features up to type-$L$, increases as $L$ grows -- especially when the feature dimension is held constant. In this study, we introduce a key lemma that allows us to analyze steerable features by examining their corresponding invariant features. The lemma facilitates us in understanding the limitations of $k$-hop invariant GNNs, which fail to capture the global geometric structure due to the loss of geometric information between local structures. Furthermore, we investigate the invariant features associated with different types of steerable features and demonstrate that the expressiveness of steerable features is primarily determined by their dimension -- independent of their irreducible decomposition. This suggests that when the feature dimension is constant, increasing $L$ does not lead to essentially improved performance in equivariant GNNs employing steerable features up to type-$L$. We substantiate our theoretical insights with numerical evidence.

Many real-world datasets are represented as 3D point clouds -- yet they often lack a predefined reference frame, posing a challenge for machine learning or general data analysis. Traditional methods for determining reference frames and normalizing 3D point clouds often struggle with specific inputs, lack theoretical guarantees, or require massive data. We introduce a new algorithm that overcomes these limitations and guarantees both universality and compatibility with any learnable framework for 3D point cloud analysis. Our algorithm works with any input point cloud and performs consistently regardless of input complexities, unlike data-driven methods that are susceptible to biases or limited training data. Empirically, our algorithm outperforms existing methods in effectiveness and generalizability across diverse benchmark datasets. Code is available at https://github.com/Utah-Math-Data-Science/alignment.

Hardware fuzzing has emerged as a promising automatic verification technique to efficiently discover and verify hardware vulnerabilities. However, hardware fuzzing can be extremely time-consuming due to compute-intensive iterative simulations. While recent research has explored several approaches to accelerate hardware fuzzing, nearly all of them are limited to single-input fuzzing using one thread of a CPU-based simulator. As a result, we propose Gen-Fuzz, a GPU-accelerated hardware fuzzer using a genetic algorithm with multiple inputs. Measuring experimental results on a real industrial design, we show that GenFuzz running on a single A6000 GPU and eight CPU cores achieves 80× runtime speed-up when compared to state-of-the-art hardware fuzzers.