Conditioning Diffusions Using Malliavin Calculus

• URL: http://arxiv.org/abs/2504.03461v1
• Date: Fri, 04 Apr 2025 14:10:21 GMT
• Title: Conditioning Diffusions Using Malliavin Calculus
• Authors: Jakiw Pidstrigach, Elizabeth Baker, Carles Domingo-Enrich, George Deligiannidis, Nikolas Nüsken,
• Abstract summary: In optimal control and conditional generative modelling, a central computational task is to modify a reference diffusion process to maximise a given terminal-time reward.<n>We introduce a novel framework, based on Malliavin calculus and path-space integration by parts, that enables the development of methods robust to such singular rewards.<n>This allows our approach to handle a broad range of applications, including classification, diffusion bridges, and conditioning without the need for artificial observational noise.
• Score: 18.62300657866048
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In stochastic optimal control and conditional generative modelling, a central computational task is to modify a reference diffusion process to maximise a given terminal-time reward. Most existing methods require this reward to be differentiable, using gradients to steer the diffusion towards favourable outcomes. However, in many practical settings, like diffusion bridges, the reward is singular, taking an infinite value if the target is hit and zero otherwise. We introduce a novel framework, based on Malliavin calculus and path-space integration by parts, that enables the development of methods robust to such singular rewards. This allows our approach to handle a broad range of applications, including classification, diffusion bridges, and conditioning without the need for artificial observational noise. We demonstrate that our approach offers stable and reliable training, outperforming existing techniques.

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