Xiao Wang

Xiao Wang, Wenpeng Shang, Xiaofan Li et al.

Non-Gaussian Lévy noises are present in many models for understanding underlining principles of physics, finance, biology and more. In this work, we consider the Fokker-Planck equation(FPE) due to one-dimensional asymmetric Lévy motion, which is a nonlocal partial differential equation. We present an accurate numerical quadrature for the singular integrals in the nonlocal FPE and develop a fast summation method to reduce the order of the complexity from $O(J^2)$ to $O(J\log J)$ in one time-step, where $J$ is the number of unknowns. We also provide conditions under which the numerical schemes satisfy maximum principle. Our numerical method is validated by comparing with exact solutions for special cases. We also discuss the properties of the probability density functions and the effects of various factors on the solutions, including the stability index, the skewness parameter, the drift term, the Gaussian and non-Gaussian noises and the domain size.

Runmin Jiang, Genpei Zhang, Yuntian Yang et al. · cmu, harvard

Single-particle cryo-electron microscopy (cryo-EM) has become a cornerstone of structural biology, enabling near-atomic resolution analysis of macromolecules through advanced computational methods. However, the development of cryo-EM processing tools is constrained by the scarcity of high-quality annotated datasets. Synthetic data generation offers a promising alternative, but existing approaches lack thorough biophysical modeling of heterogeneity and fail to reproduce the complex noise observed in real imaging. To address these limitations, we present CryoCCD, a synthesis framework that unifies versatile biophysical modeling with the first conditional cycle-consistent diffusion model tailored for cryo-EM. The biophysical engine provides multi-functional generation capabilities to capture authentic biological organization, and the diffusion model is enhanced with cycle consistency and mask-guided contrastive learning to ensure realistic noise while preserving structural fidelity. Extensive experiments demonstrate that CryoCCD generates structurally faithful micrographs, enhances particle picking and pose estimation, as well as achieves superior performance over state-of-the-art baselines, while also generalizing effectively to held-out protein families.

Daniel Keysers, Nathanael Schärli, Nathan Scales et al.

State-of-the-art machine learning methods exhibit limited compositional generalization. At the same time, there is a lack of realistic benchmarks that comprehensively measure this ability, which makes it challenging to find and evaluate improvements. We introduce a novel method to systematically construct such benchmarks by maximizing compound divergence while guaranteeing a small atom divergence between train and test sets, and we quantitatively compare this method to other approaches for creating compositional generalization benchmarks. We present a large and realistic natural language question answering dataset that is constructed according to this method, and we use it to analyze the compositional generalization ability of three machine learning architectures. We find that they fail to generalize compositionally and that there is a surprisingly strong negative correlation between compound divergence and accuracy. We also demonstrate how our method can be used to create new compositionality benchmarks on top of the existing SCAN dataset, which confirms these findings.

Yanghong Huang, Xiao Wang

Several finite difference methods are proposed for the infinitesimal generator of 1D asymmetric $α$-stable Lévy motions, based on the fact that the operator becomes a multiplier in the spectral space. These methods take the general form of a discrete convolution, and the coefficients (or the weights) in the convolution are chosen to approximate the exact multiplier after appropriate transform. The accuracy and the associated advantages/disadvantages are also discussed, providing some guidance on the choice of the right scheme for practical problems, like in the calculation of mean exit time for random processes governed by general asymmetric $α$-stable motions.