Efficient Temporal Consistency in Diffusion-Based Video Editing with Adaptor Modules: A Theoretical Framework

Adapter-based methods are commonly used to enhance model performance with minimal additional complexity, especially in video editing tasks that require frame-to-frame consistency. By inserting small, learnable modules into pretrained diffusion models, these adapters can maintain temporal coherence without extensive retraining. Approaches that incorporate prompt learning with both shared and frame-specific tokens are particularly effective in preserving continuity across frames at low training cost. In this work, we want to provide a general theoretical framework for adapters that maintain frame consistency in DDIM-based models under a temporal consistency loss. First, we prove that the temporal consistency objective is differentiable under bounded feature norms, and we establish a Lipschitz bound on its gradient. Second, we show that gradient descent on this objective decreases the loss monotonically and converges to a local minimum if the learning rate is within an appropriate range. Finally, we analyze the stability of modules in the DDIM inversion procedure, showing that the associated error remains controlled. These theoretical findings will reinforce the reliability of diffusion-based video editing methods that rely on adapter strategies and provide theoretical insights in video generation tasks.