cs.LGMar 30, 2026
SaveFrom Independent to Correlated Diffusion: Generalized Generative Modeling with Probabilistic Computers
Organizations: Department of Electrical and Computer Engineering, University of California, Santa Barbara, Santa Barbara, CA, 93106, USA
Abstract
Diffusion models have emerged as a powerful framework for generative tasks in deep learning. They decompose generative modeling into two computational primitives: deterministic neural-network evaluation and stochastic sampling. Current implementations usually place most computation in the neural network, but diffusion as a framework allows a broader range of choices for the stochastic transition kernel. Here, we generalize the stochastic sampling component by replacing independent noise injection with Markov chain Monte Carlo (MCMC) dynamics that incorporate known interaction structure. Standard independent diffusion is recovered as a special case when couplings are set to zero. By explicitly incorporating Ising couplings into the diffusion dynamics, the noising and denoising processes exploit spatial correlations representative of the target system. The resulting framework maps naturally onto probabilistic computers (p-computers) built from probabilistic bits (p-bits), which provide orders-of-magnitude advantages in sampling throughput and energy efficiency over GPUs. We demonstrate the approach on equilibrium states of the 2D ferromagnetic Ising model and the 3D Edwards-Anderson spin glass, showing that correlated diffusion produces samples in closer agreement with MCMC reference distributions than independent diffusion. More broadly, the framework shows that p-computers can enable new classes of diffusion algorithms that exploit structured probabilistic sampling for generative modeling.
Figures & tables
Fig. 1 : Hybrid p-computer/GPU inference loop for correlated diffusion. System-level view of one reverse-diffusion iteration. The current noisy configuration ( ) is passed from the probabilistic computer to the neural network on the GPU, which produces per-site denoising probabilities . A binary clean-state estimate is then sampled from these probabilities and sent back to the probabilistic computer, where it initializes an ensemble of forward Gibbs chains under the known couplings and the scheduled inverse temperatures. Each chain produces a candidate for , and the candidates are reweighted by the one-step likelihood before one is sampled to continue the reverse trajectory. Repeating this procedure produces successive reverse states , yielding the final sample . The two blocks represent a conceptual division of labor between estimation and sampling, in an integrated CMOS+sMTJ realization both can reside on the same chip.
Fig. 2 : Probabilistic computing hardware primitives. (a) p-bits and g-bits serve as the building blocks for Bernoulli and Gaussian sampling, respectively. (b) Multi-state Potts variables with integer states in can be realized either as native Potts units or as networks of interconnected p-bits. (c) A p-bit circuit and candidate stochastic nanodevice realizations, including sMTJs, Zener diodes, and SPADs. (d) Three tunable g-bits implemented on FPGA, each with 256 states and independently programmable means and variances.
| Platform | Throughput (Gsample/s) | Power (W) | Efficiency (Gsamples/J) | Improvement factor |
|---|---|---|---|---|
| GPU (experimental) | ||||
| FPGA (experimental) | ||||
| sMTJ (projected) |
Table 1: Comparison of efficiency for generating 32-bit random-number samples across hardware platforms
Fig. 3 : Independent and correlated diffusion noising in a p-bit framework: Illustration of the forward noising process implemented using probabilistic bits (p-bits), showing how structure-aware correlated noising generalizes the independent diffusion for a 2D ferromagnetic Ising system. (a) Independent diffusion noising : each p-bit is updated with under a time-dependent local bias , so that ; a decreasing inverse temperature schedule (Eq. 1 ) progressively destroys correlations, resulting in factorized, site-wise noise injection (top row). (b) Correlated diffusion noising : each p-bit is updated from an interaction-dependent local field , where explicit couplings introduce spatially correlated noise consistent with the underlying Ising structure (bottom row). The independent noising process in (a) is recovered as a special case of (b) by setting and absorbing the noising strength into the bias term .
Fig. 4 : Reverse generative process (diffusion inference) using neural-network guidance and p-bit sampling: Schematic of one reverse step. Given the current noisy state , a neural network computes per-site denoising probabilities , from which a clean-state estimate is sampled. An ensemble of forward Gibbs chains initialized at then produces candidates for on the probabilistic computer. These candidates are reweighted according to , and one candidate is sampled to continue the reverse trajectory. Repeating this procedure yields a sequence of progressively denoised states ending at .
Fig. 5 : Reverse trajectories for independent and correlated kernels on the 2D ferromagnetic Ising model. Representative configurations (right-to-left) at diffusion timesteps (red/blue denote ). Top: independent (site-wise) kernel. Middle: Gibbs-sampled reference at matching timesteps. Bottom: correlated (interaction-aware) kernel. The mean magnetization and mean energy per spin , averaged over the ensemble at each timestep, are reported beneath the corresponding column. The correlated kernel tracks the Gibbs reference more closely across timesteps, while the independent kernel produces spatially fragmented intermediate states and can deviate in magnetization even when energy appears comparable.
Fig. 6 : Correlated diffusion applied to a 3D Edwards-Anderson spin glass near the phase transition. (a) Energy-per-spin distribution of the generated samples (blue) against the Monte Carlo reference (black). (b) Training and validation loss of the conditional estimator. (c) Parisi overlap distribution of the generated samples (blue) compared with the equilibrium reference (black), showing that the framework captures the bimodal overlap structure of the spin-glass phase. (d) Mean energy per spin during the forward and reverse processes, for the generated trajectory (blue) and the Monte Carlo reference (black).
| Architecture | Training | ||
|---|---|---|---|
| Input / output size | Loss | Binary cross-entropy | |
| Hidden layers | 2 | Optimizer | Adam |
| Hidden units per layer | 1024 | Learning rate | |
| Activation | ReLU | Weight decay | None |
| Dropout | None | Batch size | 512 |
| Layer normalization | None | Epochs | 4500 |
Table S2: Architecture and training hyperparameters for the conditional estimator , used identically for the 2D ferromagnetic Ising model and the 3D spin glass.
Fig. S1 : Energy and magnetization across diffusion timesteps for generated Ising states. Mean energy per spin and absolute mean magnetization versus diffusion timestep . (a,b) Independent noising: reverse-generated states (red) are compared with forward samples from the factorized noising kernel (black). Energy appears broadly similar under a chosen schedule (a), but magnetization differs strongly (b), showing that independent noising suppresses long-range order too rapidly. (c,d) Correlated noising: reverse-generated states (blue) are compared with interaction-respecting MCMC reference states (black). Both and track the reference more closely over , consistent with the preservation of spatial correlations in the forward kernel.
Fig. S2 : Connected two-point correlation of the final generated samples ( ). for the generated ensembles of the correlated (blue) and independent (red) kernels on a ferromagnetic Ising model with open boundary conditions near the critical inverse temperature, compared against the ideal MCMC reference (black). Both kernels share the same network architecture and training protocol. The correlated kernel closely follows the reference, while the independent kernel deviates substantially. Error bars, where visible, denote the statistical uncertainty of the generated ensembles.
Fig. S3 : Parisi overlap distribution on the 3D Edwards-Anderson instance: correlated versus independent diffusion. of the generated ensembles for correlated diffusion (blue) and for an independent-diffusion baseline with the couplings removed from the diffusion kernels, (red), compared against the equilibrium MCMC reference (black), for the instance at . The two arms share the same estimator architecture, training protocol, reverse sampler, and training data, and differ only in whether the couplings enter the diffusion kernels. Neither reproduces the reference, but the correlated kernel retains weight at intermediate overlap that the independent baseline does not.
Fig. S4 : Energy autocorrelation of the reference Monte Carlo dynamics. Normalized autocorrelation of the energy per spin, , against lag in Monte Carlo sweeps, for the 2D ferromagnet at (left) and the 3D Edwards-Anderson instance at (right).