Harmonic Path Integral Diffusion

Abstract

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.