Sampling in discrete spaces, with critical applications in simulation and opti-1 mization, has recently been boosted by significant advances in gradient-based2 approaches that exploit modern accelerators like GPUs. However, two key chal-3 lenges hinder the further research progress in discrete sampling. First, since there4 is no consensus on experimental settings, the empirical results in different research5 papers are often not comparable. Secondly, implementing samplers and target6 distributions often requires a nontrivial amount of effort in terms of calibration,7 parallelism, and evaluation. To tackle these challenges, we propose DISCS (DIS-8 Crete Sampling), a tailored package and benchmark that supports unified and9 efficient implementation and evaluations for discrete sampling in three types of10 tasks: sampling for classical graphical models, combinatorial optimization, and11 energy based generative models. Throughout the comprehensive evaluations in12 DISCS, we acquired new insights into scalability, design principles for proposal13 distributions, and lessons for adaptive sampling design.

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.

Sampling in discrete spaces, with critical applications in simulation and optimization, has recently been boosted by significant advances in gradient-based approaches that exploit modern accelerators like GPUs. However, two key challenges are hindering further advancement in research on discrete sampling.

Structural connectomes are detailed graphs that map how different brain regions are physically connected, offering critical insight into aging, cognition, and neurodegenerative diseases. However, these connectomes are high-dimensional and densely interconnected, which makes them difficult to interpret and analyze at scale. While low-dimensional spaces like PCA and autoencoders are often used to capture major sources of variation, their latent spaces are generally continuous and cannot fully reflect the mixed nature of variability in connectomes, which include both continuous (e.g., connectivity strength) and discrete factors (e.g., imaging site). Motivated by this, we propose a variational autoencoder (VAE) with a hybrid latent space that jointly models the discrete and continuous components. We analyze a large dataset of 5,761 connectomes from six Alzheimer's disease studies with ten acquisition protocols. Each connectome represents a single scan from a unique subject (3579 females, 2182 males), aged 22 to 102, with 4338 cognitively normal, 809 with mild cognitive impairment (MCI), and 614 with Alzheimer's disease (AD). Each connectome contains 121 brain regions defined by the BrainCOLOR atlas. We train our hybrid VAE in an unsupervised way and characterize what each latent component captures. We find that the discrete space is particularly effective at capturing subtle site-related differences, achieving an Adjusted Rand Index (ARI) of 0.65 with site labels, significantly outperforming PCA and a standard VAE followed by clustering (p < 0.05). These results demonstrate that the hybrid latent space can disentangle distinct sources of variability in connectomes in an unsupervised manner, offering potential for large-scale connectome analysis.

However, training EBMs on data in discrete or mixed state spaces poses significant challenges due to the lack of robust and fast sampling methods. In this work, we propose to train discrete EBMs with Energy Discrepancy, a loss function which only requires the evaluation of the energy function at data points and their perturbed counterparts, thus eliminating the need for Markov chain Monte Carlo.