Harmonic Sampling: Zero-Parameter Flows on Compact Manifolds

Sampling from probability distributions over compact Riemannian manifolds is a central challenge in the natural sciences, with critical applications ranging from directional statistics to lattice quantum field theory (QFT). Recent approaches rely on manifold continuous normalizing flows to model these densities; however, training these manifold flows via likelihood maximization or distribution matching is notoriously unstable and computationally expensive. In this paper, we bypass the optimization paradigm entirely by introducing Harmonic Sampling, a novel, zero-parameter flow-based sampling framework. By projecting the target density onto the manifold's harmonic basis (the eigenfunctions of the Laplace-Beltrami operator), we analytically derive a deterministic, closed-form vector field that exactly transports a uniform prior to the target distribution. We provide rigorous theoretical guarantees demonstrating that this vector field is globally well-posed, singularity-free, and bounded, with approximation errors that decay super-algebraically. Empirically, we validate our framework on hyperspheres ($S^2$, $S^3$) and the special unitary groups $SU(2)$ and $SU(3)$. Across all evaluated geometries, our generated samples demonstrate rapid spectral convergence, recovering exact ground-truth densities. For lattice QFT applications, we show how one can easily adapt Harmonic Sampling to natively enforce strict gauge (conjugation) invariances. In short, our paper provides a method to obtain exact, high-fidelity samples given only a density function over a compact manifold, entirely eliminating the need for neural network training in this context.