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

TL;DR

This work provides a formal theoretical foundation for adapter-based diffusion video editing with temporal consistency losses in DDIM-based frameworks. It proves differentiability and Lipschitz continuity of the temporal loss, establishes convergence guarantees for gradient-based optimization, and demonstrates stability of DDIM inversion when combined with bilateral filtering. The analysis extends to token-based adapters, showing that sufficiently rich shared and unshared tokens enable near-perfect semantic alignment via cross-attention. Empirical results corroborate the theory, showing improved temporal coherence and stable frame-to-frame consistency with modest adapter overhead, making diffusion-based video editing more reliable and scalable.

Abstract

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.

Figures (4)

• Figure 1: An overview of the typical video generation process using LoRA-enhanced feature extraction. Depth and text embeddings are combined with latent vectors and processed through iterative denoising, cross attention, and cosine similarity constraints between adjacent frames.
• Figure 2: Typical adapters mechanism regarding shared and unshared token mechanism for video generation. Shared tokens ensure global consistency across frames, while unshared tokens handle frame-specific details. A same prefix is applied across all time steps, and only shared tokens are updated during the final phase.
• Figure 3: Comparison of single-channel feature heatmaps from the cross-attention layers (UNet blocks 4–11), illustrating the impact of adapter fine-tuning on attention alignment in scenarios “Two People in Conversation” and “Making Tomato Gumbo in the Kitchen.” Labels f1/f2 indicate adjacent frames, and t1/t2 represent diffusion timesteps (t1=932, t2=941). These empirical results visually confirm Theorem \ref{['theorem:attention_alignment']} and Corollary \ref{['cor:token_sufficiency']}, demonstrating that enriching token embeddings through adapter fine-tuning effectively reduces the alignment error $\|\Delta Z\|_F$, leading to precise semantic alignment and enhanced temporal consistency.
• Figure 4: The left panel presents the empirical validation of Theorem \ref{['thm:temporal_consistency']}, showing the average cosine similarity between latent representations of consecutive frames across different DDIM timesteps. With the adapter enabled, the similarity rapidly increases, approaching unity, confirming the theoretical prediction that temporal consistency improves when optimizing the temporal consistency loss. The right panel further verifies this by measuring the variation in inter-frame similarity across training epochs. Initially exhibiting substantial fluctuations without the adapter, the introduction of the adapter stabilizes these variations significantly, aligning well with the theoretical guarantee of gradient boundedness and Lipschitz continuity proven in Lemma \ref{['lemma:differentiability_sim']}.

Theorems & Definitions (26)

• Theorem 4.1: Optimizability of Temporal Consistency Loss
• Lemma 4.2: Differentiability of the Cosine Similarity
• Lemma 4.3: Lipschitz Continuity of the $\mathcal{L}_{\mathrm{temporal}}$ Gradient
• Theorem 4.4: Convergence of Gradient Descent
• Lemma 4.5: Convexity of the Temporal Consistency Loss
• proof
• Theorem 4.6: Stability of Bilateral Filtering DDIM Inversion
• Lemma 4.7: Error Contraction by Bilateral Filtering
• Lemma 4.8: Error Propagation in a Single DDIM Step
• Lemma 4.9: Expected Error Control in DDIM with Bilateral Filtering