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Discrete Diffusion Models via Evolving Variational Autoregressive Networks

Kewen Pan, Ying Tang

arXiv:2609.27306v1cs.LGcond-mat.dis-nnstat.ML

TL;DR

Score-based discrete diffusion models learn unnormalized densities, while tensor-network representations of normalized distributions are constrained by lattice dimensionality. This paper uses evolving variational autoregressive networks with explicit Markov transitions; it reports accurate Ising thermodynamic estimates and integrates the model with adaptive MCMC sampling.

  • Problem

    Score-based models represent only unnormalized densities, while tensor-network approaches to normalized discrete diffusion are constrained by lattice dimension.

    Source: §I · p. 1

  • Method

    The paper parameterizes normalized distributions at successive diffusion times with VAN and combines the discrete diffusion model with adaptive MCMC proposal updates.

    Source: §I · pp. 1–2, §IV · p. 8

  • Results

    VAN predicts magnetization more accurately than tensor networks on tested 2D lattices while giving reliable energy estimates, and the framework is applied to thermodynamic observables in 2D and 3D Ising models.

    Source: §III · pp. 5–6

  • Takeaways & Limitations

    The framework extends normalized discrete diffusion to higher-dimensional spin lattices and provides connected and disconnected MCMC updates intended to alleviate local trapping.

    Source: §V · p. 9

    Sampling reliability depends on network training, each iteration costs more than standard spin-flip updates, and strict theoretical bounds for the full MCMC process remain challenging.

    Source: §IV · p. 9

Abstract

from arXiv · show

Conventional score-based diffusion models learn scores without representing normalized densities, whereas tractable normalized models support both sampling and direct likelihood evaluation. A recent tensor-network approach provides such a representation but is largely restricted to low-dimensional lattices. Here we introduce a discrete diffusion model that parameterizes normalized probability distributions using variational autoregressive networks. Explicit Markov jump operators govern the forward noising and reverse denoising dynamics, extending discrete diffusion models with normalized distributions to spin systems on higher-dimensional lattices. We apply this framework to the two- and three-dimensional Ising models across ordered, critical, and disordered regimes, accurately computing thermodynamic quantities including free energy, energy, and magnetization. We further integrate the framework with Monte Carlo sampling, using adaptive diffusion steps to maintain high acceptance rates even at low temperatures while enhancing sample diversity. These results establish a neural-network framework for the discrete diffusion model with normalized probability distributions.

I. INTRODUCTION

Score-based discrete diffusion models represent unnormalized densities, complicating distribution-wide learning and likelihood evaluation, while tensor-network approaches face dimensionality limits. The paper replaces tensor networks with variational autoregressive networks (VAN) to retain normalized distributions across lattice geometries.

I. INTRODUCTION

  • Score-based models represent only unnormalized densities, making full-support learning difficult and likelihood—and therefore free-energy—evaluation non-trivial.Finite-time denoising also introduces reconstruction error when the forward process has not reached its noise prior.Source: §I · p. 1
  • Tensor-network diffusion provides normalized distributions and exact likelihoods, but its MPS representation is suited to quasi-one-dimensional geometries and becomes prohibitively costly as spatial dimension grows.On 8 × 30 and 9 × 30 cylindrical Ising lattices, magnetization already deviates from Monte Carlo benchmarks at criticality; direct 3D application is infeasible.Source: §I · p. 1
  • The paper replaces tensor networks with VAN, which parameterizes normalized distributions at each diffusion step under fixed transition operators.The autoregressive ansatz is used in place of a matrix product state.Source: §I · p. 1, Figure 1 · p. 2
  • VAN makes likelihood available at every diffusion step for direct variational free-energy evaluation and applies across 2D and 3D lattices without a bond dimension.The framework is also integrated with MCMC to generate non-local updates, sustain high acceptance at low temperatures, and enhance sample diversity.Source: §I · p. 2

A. Variational autoregressive networks

The variational autoregressive network approximates the Boltzmann distribution through a tractable factorization of configuration probabilities. Its autoregressive structure supports likelihood evaluation and sampling, while reverse-KL training fits the target distribution.

A. Variational autoregressive networks

  • The configuration space contains 2^D spin states, so exact computation of the partition function becomes intractable as system size grows.The target distribution is the Boltzmann distribution, with inverse temperature β = 1/T and energy E(σ).Source: A. Variational autoregressive networks · p. 2
  • VAN makes modeling tractable by factorizing the joint distribution into sequential conditional probabilities.The factorization follows a prescribed order.Source: A. Variational autoregressive networks · p. 2
  • The autoregressive structure captures long-range spin correlations and enables precise likelihood evaluation and sequential configuration sampling.Generated configurations can be sampled independently in the predefined order.Source: A. Variational autoregressive networks · p. 3
  • Training minimizes reverse KL divergence between the parameterized VAN and target Boltzmann distribution, using samples drawn from the model and explicit energy information.The variational free energy upper-bounds the exact free energy; minimizing it narrows the gap and drives the model distribution toward the target.Source: A. Variational autoregressive networks · p. 3
  • VAN's geometry-agnostic neural-network ansatz supports arbitrary topologies, including 2D convolutional modeling, and is incorporated into discrete diffusion to approximate normalized probabilities over time.The paper situates this dynamic use within VAN's application to nonequilibrium statistical mechanics.Source: A. Variational autoregressive networks · pp. 3–4

B. Discrete noising process

The forward noising process is a continuous-time Markov chain on discrete configurations. Single-spin-flip transitions define a bistochastic generator, and a first-order Euler approximation yields tractable finite-step evolution.

B. Discrete noising process

  • The noising dynamics form a continuous-time Markov chain whose probability distribution evolves according to a master equation.The evolution between times t1 and t2 can be expressed using the Markov generator.Source: B. Discrete noising process · p. 4
  • The bistochastic noising process allows only transitions between configurations differing by one spin flip and preserves normalization while converging asymptotically to the uniform distribution.The generator assigns transition rates to these single-spin changes.Source: B. Discrete noising process · p. 4
  • For sufficiently small Δt, the transition matrix uses the first-order Euler approximation e^(ΔtW) ≈ I + ΔtW, with DΔt ≤ 1.This approximation constructs the forward noising process governed by the single-spin-flip generator.Source: B. Discrete noising process · p. 4

C. Discrete denoising process

The model reverses the Euler-discretized noising process using Bayes-derived updates, with a stepwise scheme for accuracy and tau-leaping for faster approximate denoising. Tau-leaping trades strict numerical accuracy for lower computational overhead.

C. Discrete denoising process

  • The Euler-discretized reverse process is derived analytically from Bayes' theorem to simulate probability flow between adjacent time steps.The stepwise algorithm initializes a noisy state and iteratively reconstructs latent states until returning a sample at t = 0.Source: C. Discrete denoising process · pp. 4–5
  • Stepwise denoising updates the state across each time interval using the noising rule, retaining a positive probability of leaving the state unchanged.The implementation draws an initial state from the model at the terminal time and iterates backward through the time partition.Source: C. Discrete denoising process · pp. 4–5
  • Long stepwise trajectories incur substantial computational overhead and low sampling efficiency, motivating tau-leaping to accelerate reverse-process simulation.Tau-leaping bundles multiple transition events into a step, unlike the single-event Gillespie algorithm.Source: C. Discrete denoising process · p. 5
  • Tau-leaping holds reverse rates and state constant over duration τ, samples Poisson transition counts, aggregates them, and projects the result back to the valid discrete state space.It reduces computational cost, while stepwise Euler denoising is retained when strict numerical accuracy is required.Source: C. Discrete denoising process · p. 5

III. NUMERICAL RESULTS

The VAN method is more accurate than tensor networks for 2D magnetization and gives reliable energy estimates; broader 2D and 3D simulations evaluate thermodynamic observables across temperature regimes.

III. NUMERICAL RESULTS

  • The 2D Ising setup uses nearest-neighbor interactions with J = 1, and its ordered and disordered regimes lie above and below βc, respectively.The transition is continuous in the thermodynamic limit.Source: §III · p. 5
  • At the 2D critical inverse temperature, the study uses cylindrical boundaries on 8 × 30 and 9 × 30 lattices, where tensor networks show numerical bias against Monte Carlo benchmarks.The lattices use periodic boundaries along the first dimension and open boundaries along the second.Source: §III · p. 5
  • VAN predicts 2D magnetization per site more accurately than tensor networks and gives reliable energy estimates in the 8 × 30 and 9 × 30 comparisons.Wolff cluster algorithm results serve as reference data; tensor-network energy data are unavailable in the cited reference.Source: §III · p. 5, Figure 3 · p. 6
  • The 16 × 16 2D and 4 × 4 × 4 3D simulations compute free energy, energy, and magnetization per site across ordered, disordered, and critical regimes.FlashVAN values benchmark free energy because the 3D Ising model has no exact solution; tensor-network cost and memory hinder accurate characterization of long-range correlations in 3D.Source: §III · pp. 5–6

IV. INTEGRATING DISCRETE DIFFUSION INTO MCMC PROPOSAL UPDATES

The paper embeds discrete-diffusion proposals in an MCMC sampler, using noising and denoising to explore spin configurations beyond local updates. In Ising-model quenches, the proposed methods outperform local Metropolis sampling on acceptance rates, while trading greater per-iteration cost for broader exploration.

IV. INTEGRATING DISCRETE DIFFUSION INTO MCMC PROPOSAL UPDATES

  • The DDM proposal module is embedded in MCMC to sample the target distribution rather than relying solely on a learned distribution that may mismatch it.The paper identifies mode collapse, biased results, and poor sample diversity as possible symptoms of distribution mismatch.Source: §IV · p. 6
  • Training first fits a VAN to the target distribution, then iteratively trains each forward-diffusion step using supervised targets computed from the transition rate and time step.For later steps, the preceding step’s trained parameters initialize the VAN, which is fine-tuned.Source: §IV · p. 6
  • MCMC chains start from autoregressive VAN samples, then repeatedly use their accepted configurations as the next iteration’s initial states until convergence.Each iteration corrupts configurations and reverses the process to form candidates, which are accepted or rejected by Metropolis–Hastings.Source: §IV · pp. 6–7
  • Larger temperature gaps reduce both acceptance and decorrelation, while more diffusion steps lower acceptance but improve decorrelation; the sampler adjusts diffusion time against an acceptance target.The adaptive rule increases diffusion time when acceptance exceeds the preset threshold and decreases it otherwise.Source: §IV · pp. 7–8, Figure 5 · p. 8
  • Forward random spin flips diversify configurations, and reverse denoising restricts flips to guide proposals toward the learned distribution, producing non-local updates beyond LMMC’s locality bottleneck.The paper says this denoising can moderately increase subsequent Metropolis–Hastings acceptance by mitigating large deviations from the target distribution.Source: §IV · p. 8
  • The paper compares connected DDM updates with disconnected NMCMC updates, where the latter proposes a configuration statistically independent of the current state.The two proposal schemes are accepted or rejected using the Metropolis–Hastings procedure.Source: §IV · p. 8
  • The connected and disconnected diffusion updates sustain higher acceptance rates than LMMC in the low-temperature regime, especially the connected update, while broader exploration yields more diverse samples and fewer convergence iterations at higher per-iteration cost.The comparison uses identical runtimes and equal sample counts for diversity; the diffusion methods have higher computational cost per iteration.Source: §IV · pp. 8–9, Figure 6 · p. 9
  • Sampling reliability depends on network training, candidate generation and likelihood evaluation raise per-iteration costs, and strict theoretical bounds for the full MCMC process remain challenging.Training may miss low-probability configurations with physical significance; faster chain mixing can partly offset the added computational overhead.Source: §IV · p. 9

V. DISCUSSION

The discussion presents an evolving VAN as a normalized discrete diffusion model for higher-dimensional spin lattices, with tau-leaping and MCMC integrations. Experiments on 2D and 3D Ising models report accurate thermodynamic observables, while the authors identify extensions to more complex systems and acceleration methods as future directions.

V. DISCUSSION

  • The evolving-VAN model captures free energy, energy, and magnetization across temperatures in 2D and 3D Ising systems, with competitive or better accuracy than tensor-network baselines, especially in 3D.Tau-leaping reduces iterative computational overhead while maintaining acceptable precision in most sampling scenarios.Source: §V · p. 9
  • Connected diffusion trajectories and disconnected VAN proposals form complementary MCMC updates that alleviate local trapping in conventional LMMC methods.Connected updates create correlated candidates through forward noising and reverse denoising; disconnected updates generate independent proposals through VAN sampling.Source: §V · p. 9
  • The connected update links forward noising to sequential single-spin LMMC flips; reverse denoising is a weakened disconnected update that recovers the full update as t →∞.The paper presents this as a connection between the two sampling strategies.Source: §V · p. 10
  • The authors propose testing generalization beyond classical ferromagnetic Ising models and incorporating distillation or enhanced tau-leaping for faster denoising.They also identify stochastic-process simulation as a possible direction for quantum generative models.Source: §V · p. 10

Appendix A: Notation

The appendix points readers to a table of mathematical notation and describes a MADE implementation whose fixed masks enforce autoregressive spin dependencies. It also outlines network settings and a tau-leaping validation procedure for 2D and 3D Ising models.

Appendix A: Notation

  • Table I is identified as the location of the paper’s mathematical-symbol explanations.The supplied passage does not specify individual symbol definitions.Source: Appendix A · p. 10, Table I · p. 10

Appendix B: Masked autoencoder for distribution estimation

  • Fixed block-wise binary masks applied elementwise to layer weights enforce autoregressive constraints by blocking redundant spin connections.The input projection uses an exclusive strictly lower-triangular mask to exclude self-dependency; remaining layers use inclusive lower-triangular masks, preserving sequential order.Source: Appendix B · pp. 10–11
  • Figure 7 compares post-denoising energy and magnetization with initial-sample baselines across tau-leap steps for 2D and 3D Ising models.The selected leap duration is the step where solid curves and dashed baselines agree most closely.Source: Figure 7 · p. 11
  • The 2D model uses depth 3 and width 32, while the 3D model uses depth 4 and width 16; Adam and natural gradient descent are both supported.Typical learning rates are 10^-4 or 10^-5 for Adam and 10^-2 for natural gradient descent.Source: Appendix B · p. 11

Appendix C: Optimal leap duration for tau-leaping

The appendix describes how leap duration controls the precision–efficiency trade-off in denoising simulations. It notes that unsuitable durations can reduce accuracy and that Euler-discretized denoising is an alternative when no acceptable leap duration is found.

Appendix C: Optimal leap duration for tau-leaping

  • Tau-leaping’s duration controls the trade-off between numerical precision and computational efficiency in denoising simulations.The forward dynamics use fixed-size jump intervals, with learned transition rates held constant within each interval.Source: Appendix C · p. 11
  • A greater deviation between the adopted τ and the optimal leap duration leads to lower denoising accuracy.The passage relates this sensitivity to the noising process’s Euler approximation fitting.Source: Appendix C · p. 11
  • When no suitable τ keeps numerical errors acceptable in high-precision scenarios, Euler-discretized denoising is a reliable alternative.Source: Appendix C · p. 11

Appendix D: Sampling algorithm

The appendix describes the Wolff cluster sampler and its use as a baseline for energy and magnetization. It also notes that its cluster construction is tailored to particular interaction structures, limiting its applicability to generic disordered or asymmetric systems.

Appendix D: Sampling algorithm

  • Wolff cluster sampling is used to compute baseline mean energy and magnetization per site, and its single-cluster updates suppress critical slowing down near second-order transitions.The method is distinguished from Swendsen–Wang by its single-cluster update.Source: Appendix D · p. 11
  • Wolff’s cluster-building principle is tailored to ferromagnets with uniform short-range couplings, while its reliance on structured interactions limits adaptability to generic disordered or asymmetric systems.Source: Appendix D · pp. 11–12
  • The algorithm starts from a uniformly selected lattice site, grows a cluster using a stack, and flips all spins in that cluster.Growth examines nearest neighbors and considers neighbors with the same spin as the current site.Source: Appendix D · p. 12
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