Beyond First-Order Tweedie: Solving Inverse Problems using Latent Diffusion (2312.00852v1)

Abstract: Sampling from the posterior distribution poses a major computational challenge in solving inverse problems using latent diffusion models. Common methods rely on Tweedie's first-order moments, which are known to induce a quality-limiting bias. Existing second-order approximations are impractical due to prohibitive computational costs, making standard reverse diffusion processes intractable for posterior sampling. This paper introduces Second-order Tweedie sampler from Surrogate Loss (STSL), a novel sampler that offers efficiency comparable to first-order Tweedie with a tractable reverse process using second-order approximation. Our theoretical results reveal that the second-order approximation is lower bounded by our surrogate loss that only requires $O(1)$ compute using the trace of the Hessian, and by the lower bound we derive a new drift term to make the reverse process tractable. Our method surpasses SoTA solvers PSLD and P2L, achieving 4X and 8X reduction in neural function evaluations, respectively, while notably enhancing sampling quality on FFHQ, ImageNet, and COCO benchmarks. In addition, we show STSL extends to text-guided image editing and addresses residual distortions present from corrupted images in leading text-guided image editing methods. To our best knowledge, this is the first work to offer an efficient second-order approximation in solving inverse problems using latent diffusion and editing real-world images with corruptions.

• The paper introduces STSL, a second-order Tweedie sampler that improves posterior sampling in latent diffusion models.
• STSL reduces neural function evaluations by 4X-8X compared to state-of-the-art solvers, excelling in tasks such as deblurring, super-resolution, and inpainting.
• The surrogate loss framework using the trace of the Hessian provides a tractable approach that elevates image editing and restoration quality.

Overview of "Beyond First-Order Tweedie: Solving Inverse Problems using Latent Diffusion"

This paper introduces a novel approach to solving inverse problems with latent diffusion models by proposing the Second-order Tweedie Sampler from Surrogate Loss (STSL). The focus is on improving the efficiency and quality of posterior sampling processes, aiming to circumvent the limitations associated with first-order Tweedie approximations, which traditionally induce a quality-limiting bias. STSL leverages second-order approximation methods and a surrogate loss function to ensure a tractable reverse diffusion process, enhancing both computational efficiency and sampling quality.

Key Contributions

1. Second-order Tweedie Approximation: The authors develop a sampler that employs the second-order Tweedie approximation while maintaining computational efficiency similar to first-order models. This is achieved without incurring excessive time and memory costs or making the reverse diffusion process intractable for posterior sampling.
2. Algorithmic Efficiency: The STSL algorithm significantly reduces the number of neural function evaluations (NFEs) required, outperforming state-of-the-art (SoTA) solvers such as PSLD and P2L. Specifically, STSL demonstrates a 4X and 8X reduction in NFEs, respectively, while delivering superior sampling quality on datasets like FFHQ, ImageNet, and COCO.
3. Versatility in Inversion and Editing Tasks: STSL shows exemplary performance across various inversion tasks such as motion deblurring, super-resolution, and inpainting. Additionally, it extends to text-guided image editing where it effectively handles image corruptions, a domain where many existing methods underperform.
4. Surrogate Loss Framework: The theoretical foundation of STSL is built on a lower-bound derivation for second-order approximation, simplifying the computational requirements to using the trace of the Hessian.

Practical and Theoretical Implications

Practically, STSL marks a significant leap in efficiency for high-dimensional inverse problem-solving with large-scale pre-trained models like Stable Diffusion, offering practical runtime reductions and high-quality image restorations even in complex settings. Theoretically, it opens new avenues for approximation strategies in diffusion-based generative models, challenging the status quo of first-order dependence which often leads to undesired biases.

The work also highlights the potential for future exploration in the field of posterior sampling, emphasizing the integration of efficient second-order estimates in solving real-world image corruptions and maintaining content fidelity during image editing tasks.

Speculations for Future AI Developments

Challenges addressed by STSL in inverse problems could shape future developments in AI by inspiring the creation of more sophisticated approximation models. The proposed trace-based surrogate loss function might find uses beyond current applications, potentially enhancing other model domains that require efficient posterior analysis or detail-oriented image manipulations.

Furthermore, as AI systems become more integral in photorealistic image synthesis and editing, the principles and methodologies advocated in this paper could guide framework developments that balance efficiency, quality, and generalizability, thereby broadening the accessibility and applicability of advanced image processing capabilities across diverse fields.

In essence, "Beyond First-Order Tweedie" encapsulates a strategic shift in diffusion model applications, bridging theoretical advancements with practical AI solutions in image inversion and editing realms.