Sampling with Shielded Langevin Monte Carlo Using Navigation Potentials

TL;DR

This paper addresses constrained sampling for distributions defined on punctured domains, where the feasible set is non-convex due to convex obstacles: $p_F(x) \propto p(x) \mathbb{I}\{x \notin \mathcal{C}\}$ with $p(x) \propto e^{-U(x)}$. It introduces shielded Langevin Monte Carlo (shielded LMC), which blends the Rimon-Koditschek navigation potential $\phi_\alpha$ with a state-dependent diffusion (adaptive temperature) to enforce obstacle avoidance while preserving exploration, yielding the update $x_{k+1} = x_k + \eta_k[ \beta(x_k) \nabla \log p(x_k) - (\log p(x_k) \nabla \beta(x_k))/\alpha ] + \sqrt{2 \eta_k \tau \beta(x_k)^2} w_k$. The approach is validated on a 2D Gaussian mixture with circular obstacles and a MIMO symbol-detection task, showing that the sampler respects constraints, accelerates convergence compared to unconstrained Langevin, and improves downstream performance relative to unconstrained baselines. This framework enables robust constrained sampling in non-convex punctured spaces and has potential adaptations to diffusion-models and more complex obstacle geometries.

Abstract

We introduce shielded Langevin Monte Carlo (LMC), a constrained sampler inspired by navigation functions, capable of sampling from unnormalized target distributions defined over punctured supports. In other words, this approach samples from non-convex spaces defined as convex sets with convex holes. This defines a novel and challenging problem in constrained sampling. To do so, the sampler incorporates a combination of a spatially adaptive temperature and a repulsive drift to ensure that samples remain within the feasible region. Experiments on a 2D Gaussian mixture and multiple-input multiple-output (MIMO) symbol detection showcase the advantages of the proposed shielded LMC in contrast to unconstrained cases.

Figures (3)

• Figure 1: Comparison of constraint types. Left: Our setting, where convex obstacles carve out forbidden regions, yielding a non-convex feasible space from which we must sample. Right: The classical case, where sampling is restricted to a convex feasible region within the ambient space.
• Figure 2: Impact of parameter $\alpha$ on shielded LMC with obstacles. Blue dots are Langevin samples after 50k iterations, red dashed lines are obstacles, and black crosses mark mode means; contours show the target distribution.(a) $\alpha = 0.1$: repulsion is too strong, pushing samples away from true modes. (b) $\alpha = 1$: good balance between target sampling and obstacle avoidance. (c) $\alpha = 7$: repulsion is too weak, causing samples to collapse onto obstacle boundaries where gradients vanish.
• Figure 3: Performance analysis of our proposed methods for symbol detection estimation considering SER as a function of SNR in a Rayleigh fading channel model. (a) Constellation (QPSK) with the obstacles. (b) Ablation of our proposed method with respect to $\alpha$. (c) Comparison of our proposed sampler with two baselines: the unconstrained sampler and the ML, which is the optimal one.