Hamidreza Behjoo

Hamidreza Behjoo, Michael Chertkov

Computing the partition function, $Z$, of an Ising model over a graph of $N$ \enquote{spins} is most likely exponential in $N$. Efficient variational methods, such as Belief Propagation (BP) and Tree Re-Weighted (TRW) algorithms, compute $Z$ approximately by minimizing the respective (BP- or TRW-) free energy. We generalize the variational scheme by building a $λ$-fractional interpolation, $Z^{(λ)}$, where $λ=0$ and $λ=1$ correspond to TRW- and BP-approximations, respectively. This fractional scheme -- coined Fractional Belief Propagation (FBP) -- guarantees that in the attractive (ferromagnetic) case $Z^{(TRW)} \geq Z^{(λ)} \geq Z^{(BP)}$, and there exists a unique (\enquote{exact}) $λ_*$ such that $Z=Z^{(λ_*)}$. Generalizing the re-parametrization approach of \citep{wainwright_tree-based_2002} and the loop series approach of \citep{chertkov_loop_2006}, we show how to express $Z$ as a product, $\forall λ:\ Z=Z^{(λ)}{\tilde Z}^{(λ)}$, where the multiplicative correction, ${\tilde Z}^{(λ)}$, is an expectation over a node-independent probability distribution built from node-wise fractional marginals. Our theoretical analysis is complemented by extensive experiments with models from Ising ensembles over planar and random graphs of medium and large sizes. Our empirical study yields a number of interesting observations, such as the ability to estimate ${\tilde Z}^{(λ)}$ with $O(N^{2::4})$ fractional samples and suppression of variation in $λ_*$ estimates with an increase in $N$ for instances from a particular random Ising ensemble, where $[2::4]$ indicates a range from $2$ to $4$. We also discuss the applicability of this approach to the problem of image de-noising.

Hamidreza Behjoo, Michael Chertkov

In this manuscript, we present a novel approach for sampling from a continuous multivariate probability distribution, which may either be explicitly known (up to a normalization factor) or represented via empirical samples. Our method constructs a time-dependent bridge from a delta function centered at the origin of the state space at $t=0$, optimally transforming it into the target distribution at $t=1$. We formulate this as a Stochastic Optimal Control problem of the Path Integral Control type, with a cost function comprising (in its basic form) a quadratic control term, a quadratic state term, and a terminal constraint. This framework, which we refer to as Harmonic Path Integral Diffusion (H-PID), leverages an analytical solution through a mapping to an auxiliary quantum harmonic oscillator in imaginary time. The H-PID framework results in a set of efficient sampling algorithms, without the incorporation of Neural Networks. The algorithms are validated on two standard use cases: a mixture of Gaussians over a grid and images from CIFAR-10. The transparency of the method allows us to analyze the algorithms in detail, particularly revealing that the current weighted state is an order parameter for the dynamic phase transition, signaling earlier, at $t<1$, that the sample generation process is almost complete. We contrast these algorithms with other sampling methods, particularly simulated annealing and path integral sampling, highlighting their advantages in terms of analytical control, accuracy, and computational efficiency on benchmark problems. Additionally, we extend the methodology to more general cases where the underlying stochastic differential equation includes an external deterministic, possibly non-conservative force, and where the cost function incorporates a gauge potential term.

Hamidreza Behjoo, Michael Chertkov

We investigate diffusion models generating synthetic samples from the probability distribution represented by the Ground Truth (GT) samples. We focus on how GT sample information is encoded in the Score Function (SF), computed (not simulated) from the Wiener-Ito (WI) linear forward process in the artifical time $t\in [0\to \infty]$, and then used as a nonlinear drift in the simulated WI reverse process with $t\in [\infty\to 0]$. We propose U-Turn diffusion, an augmentation of a pre-trained diffusion model, which shortens the forward and reverse processes to $t\in [0\to T_u]$ and $t\in [T_u\to 0]$. The U-Turn reverse process is initialized at $T_u$ with a sample from the probability distribution of the forward process (initialized at $t=0$ with a GT sample) ensuring a detailed balance relation between the shorten forward and reverse processes. Our experiments on the class-conditioned SF of the ImageNet dataset and the multi-class, single SF of the CIFAR-10 dataset reveal a critical Memorization Time $ T_m $, beyond which generated samples diverge from the GT sample used to initialize the U-Turn scheme, and a Speciation Time $ T_s $, where for $ T_u > T_s > T_m $, samples begin representing different classes. We further examine the role of SF non-linearity through a Gaussian Test, comparing empirical and Gaussian-approximated U-Turn auto-correlation functions, and showing that the SF becomes effectively affine for $ t > T_s $, and approximately affine for $t\in [T_m,T_s]$.

Michael Chertkov, Sungsoo Ahn, Hamidreza Behjoo

In this manuscript, we introduce a novel Decision Flow (DF) framework for sampling decisions from a target distribution while incorporating additional guidance from a prior sampler. DF can be viewed as an AI-driven algorithmic reincarnation of the Markov Decision Process (MDP) approach in stochastic optimal control. It extends the continuous-space, continuous-time Path Integral Diffusion sampling technique of [Behjoo, Chertkov 2025] to discrete time and space, while also generalizing the Generative Flow Network (GFN) framework of [Bengio, et al 2021]. In its most basic form an explicit formulation that does not require Neural Networks (NNs), DF leverages the linear solvability of the underlying MDP [Todorov, 2007] to adjust the transition probabilities of the prior sampler. The resulting Markov process is expressed as a convolution of the reverse-time Green's function of the prior sampling with the target distribution. We illustrate the DF framework through an example of sampling from the Ising model -- compare DF to Metropolis-Hastings to quantify its efficiency, discuss potential NN-based extensions, and outline how DF can enhance guided sampling across various applications.

Hamidreza Behjoo, Michael Chertkov

In this study, we introduce a novel method for generating new synthetic samples that are independent and identically distributed (i.i.d.) from high-dimensional real-valued probability distributions, as defined implicitly by a set of Ground Truth (GT) samples. Central to our method is the integration of space-time mixing strategies that extend across temporal and spatial dimensions. Our methodology is underpinned by three interrelated stochastic processes designed to enable optimal transport from an easily tractable initial probability distribution to the target distribution represented by the GT samples: (a) linear processes incorporating space-time mixing that yield Gaussian conditional probability densities, (b) their diffusion bridge analogs that are conditioned to the initial and final state vectors, and (c) nonlinear stochastic processes refined through score-matching techniques. The crux of our training regime involves fine-tuning the nonlinear model, and potentially the linear models -- to align closely with the GT data. We validate the efficacy of our space-time diffusion approach with numerical experiments, laying the groundwork for more extensive future theory and experiments to fully authenticate the method, particularly providing a more efficient (possibly simulation-free) inference.

Michael Chertkov, Hamidreza Behjoo

Diffusion-based samplers -- Score Based Diffusions, Bridge Diffusions and Path Integral Diffusions -- match a target at terminal time, but the real leverage comes from choosing the schedule that governs the intermediate-time dynamics. We develop a path-wise schedule -- selection gramework for Harmonic PID with a time-varying stiffness, exploiting Piece-Wise-Constant(PWC) parametrizations and a simple hierarchical refinement. We introduce schedule-sensitive Quality-of-Sampling (QoS) diagnostics. Assuming a Gaussian-Mixture (GM) target, we retain closed-form Green functions' ration and numerically stable, Neural-Network free oracles for predicted-state maps and score. Experiments in 2D show that QoS driven PWC schedules consistently improve early-exit fidelity, tail accuracy, conditioning of the dynamics, and speciation (label-selection) timing at fixed integration budgets.