BELM: Bidirectional Explicit Linear Multi-step Sampler for Exact Inversion in Diffusion Models

The inversion of diffusion model sampling, which aims to find the corresponding initial noise of a sample, plays a critical role in various tasks.Recently, several heuristic exact inversion samplers have been proposed to address the inexact inversion issue in a training-free manner. We highlight this bidirectional explicit constraint is the key of mathematically exact inversion.We systematically investigate the Local Truncation Error (LTE) within the BELM framework and show that the existing heuristic designs of exact inversion samplers yield sub-optimal LTE.Consequently, we propose the Optimal BELM (O-BELM) sampler through the LTE minimization approach.We conduct additional analysis to substantiate the theoretical stability and global convergence property of the proposed optimal sampler.Comprehensive experiments demonstrate our O-BELM sampler establishes the exact inversion property while achieving high-quality sampling.Additional experiments in image editing and image interpolation highlight the extensive potential of applying O-BELM in varying applications.