In this manuscript we introduce a novel Decision Flow (DF) framework for sampling 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 to discrete time and space, while also generalizing the Generative Flow Network framework. In its most basic form, an explicit, Neural Network (NN) free formulation, DF leverages the linear solvability of the the underlying MDP 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, discuss potential NN based extensions, and outline how DF can enhance guided sampling across various applications.

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.

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 bridge-diffusion 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.

We present a comprehensive examination of score-based diffusion models of AI for generating synthetic images. These models hinge upon a dynamic auxiliary time mechanism driven by stochastic differential equations, wherein the score function is acquired from input images. Our investigation unveils a criterion for evaluating efficiency of the score-based diffusion models: the power of the generative process depends on the ability to de-construct fast correlations during the reverse/de-noising phase. To improve the quality of the produced synthetic images, we introduce an approach coined "U-Turn Diffusion". The U-Turn Diffusion technique starts with the standard forward diffusion process, albeit with a condensed duration compared to conventional settings. Subsequently, we execute the standard reverse dynamics, initialized with the concluding configuration from the forward process. This U-Turn Diffusion procedure, combining forward, U-turn, and reverse processes, creates a synthetic image approximating an independent and identically distributed (i.i.d.) sample from the probability distribution implicitly described via input samples. To analyze relevant time scales we employ various analytical tools, including auto-correlation analysis, weighted norm of the score-function analysis, and Kolmogorov-Smirnov Gaussianity test. The tools guide us to establishing that the Kernel Intersection Distance, a metric comparing the quality of synthetic samples with real data samples, is minimized at the optimal U-turn time.

Inference efforts -- required to compute partition function, $Z$, of an Ising model over a graph of $N$ ``spins" -- are most likely exponential in $N$. Efficient variational methods, such as Belief Propagation (BP) and Tree Re-Weighted (TRW) algorithms, compute $Z$ approximately minimizing respective (BP- or TRW-) free energy. We generalize the variational scheme building a $\lambda$-fractional-homotopy, $Z^{(\lambda)}$, where $\lambda=0$ and $\lambda=1$ correspond to TRW- and BP-approximations, respectively, and $Z^{(\lambda)}$ decreases with $\lambda$ monotonically. Moreover, this fractional scheme guarantees that in the attractive (ferromagnetic) case $Z^{(TRW)}\geq Z^{(\lambda)}\geq Z^{(BP)}$, and there exists a unique (``exact") $\lambda_*$ such that, $Z=Z^{(\lambda_*)}$. Generalizing the re-parametrization approach of \cite{wainwright_tree-based_2002} and the loop series approach of \cite{chertkov_loop_2006}, we show how to express $Z$ as a product, $\forall \lambda:\ Z=Z^{(\lambda)}{\cal Z}^{(\lambda)}$, where the multiplicative correction, ${\cal Z}^{(\lambda)}$, 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. The empirical study yields a number of interesting observations, such as (a) ability to estimate ${\cal Z}^{(\lambda)}$ with $O(N^4)$ fractional samples; (b) suppression of $\lambda_*$ fluctuations with increase in $N$ for instances from a particular random Ising ensemble.