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.

Diffusion Probabilistic Models (DPMs) have achieved considerable success in generation tasks. As sampling from DPMs is equivalent to solving diffusion SDE or ODE which is time-consuming, numerous fast sampling methods built upon improved differential equation solvers are proposed.

The sampling method alternates between adding substantial noise in additional forward steps and strictly following a backward ODE.

A.1.1 V ariance Preserving SDE as a potential function We can re-express the widely used V ariance Preserving (VP-SDE) (DDPM): dY

Generative diffusion models have recently emerged as a leading approach for generating high-dimensional data.

As a result, it tractably draws samples even when the matrices forming the Khatri-Rao product have tens of millions of rows each.

We also define the clean reversed conditional transition as Eq. Thus, a( t) and b (t) can be derived as Eq. The KL-divergence loss of the reversed transition can be simplified as Eq. Thus, we can finally write down the clean loss function Eq. (9) with reparametrization This section will further extend the derivation of the clean diffusion models in Appendix B.1 and Recall the definition of the backdoor reversed conditional transition in Eq. (10). We mark the coefficients of the r as red.

Diffusion Models (DMs) are state-of-the-art generative models that learn a reversible corruption process from iterative noise addition and denoising.