Variational Flow Maps: Make Some Noise for One-Step Conditional Generation

Flow maps enable high-quality image generation in a single forward pass. However, unlike iterative diffusion models, their lack of an explicit sampling trajectory impedes incorporating external constraints for conditional generation and solving inverse problems. We put forth Variational Flow Maps, a framework for conditional sampling that shifts the perspective of conditioning from "guiding a sampling path", to that of "learning the proper initial noise". Specifically, given an observation, we seek to learn a noise adapter model that outputs a noise distribution, so that after mapping to the data space via flow map, the samples respect the observation and data prior. To this end, we develop a principled variational objective that jointly trains the noise adapter and the flow map, improving noise-data alignment, such that sampling from complex data posterior is achieved with a simple adapter. Experiments on various inverse problems show that VFMs produce well-calibrated conditional samples in a single (or few) steps. For ImageNet, VFM attains competitive fidelity while accelerating the sampling by orders of magnitude compared to alternative iterative diffusion/flow models.

In contrast to iterative diffusion models, one/few-step generative models such as flow maps cannot naturally solve inverse problem (e.g. image de-blurring, inpainting, etc...) due to the lack of a sampling trajectory, which one can use to steer towards high-likelihood regions. To address this problem, we shifted the perspective of conditioning from "guiding a sampling path" to "finding the initial noise" that leads to high-likelihood samples. This helps to solve a wide variety of inverse problems, as well as more general reward alignment problems, while retaining the blazing fast inference speed of one/few-step flow maps.