High-dimensional and complex discrete distributions often exhibit multimodal behavior due to inherent discontinuities, posing significant challenges for sampling. Gradient-based discrete samplers, while effective, frequently become trapped in local modes when confronted with rugged or disconnected energy landscapes. This limits their ability to achieve adequate mixing and convergence in high-dimensional multimodal discrete spaces. To address these challenges, we propose \emph{Hyperbolic Secant-squared Gibbs-Sampling (HiSS)}, a novel family of sampling algorithms that integrates a \emph{Metropolis-within-Gibbs} framework to enhance mixing efficiency. HiSS leverages a logistic convolution kernel to couple the discrete sampling variable with the continuous auxiliary variable in a joint distribution. This design allows the auxiliary variable to encapsulate the true target distribution while facilitating easy transitions between distant and disconnected modes. We provide theoretical guarantees of convergence and demonstrate empirically that HiSS outperforms many popular alternatives on a wide variety of tasks, including Ising models, binary neural networks, and combinatorial optimization.

翻译：高维且复杂的离散分布常因固有非连续性呈现多模态行为，为采样带来重大挑战。基于梯度的离散采样器虽有效，但在面对崎岖或非连通的能量景观时，常陷入局部模态，从而限制了其在高维多模态离散空间中实现充分混合与收敛的能力。为解决这些问题，我们提出**双曲正割平方吉布斯采样（HiSS）**，一种新颖的采样算法族，其通过整合**Metropolis-within-Gibbs**框架来提升混合效率。HiSS利用逻辑卷积核，将离散采样变量与连续辅助变量耦合于联合分布中。该设计使得辅助变量既能封装真实目标分布，又能促进远距离且非连通的模态之间的轻松转移。我们提供了收敛性的理论保证，并通过实验证明，在伊辛模型、二元神经网络和组合优化等广泛任务中，HiSS优于许多流行替代方案。