Linear combinations of latents in generative models: subspaces and beyond (2408.08558v7)

Abstract: Sampling from generative models has become a crucial tool for applications like data synthesis and augmentation. Diffusion, Flow Matching and Continuous Normalising Flows have shown effectiveness across various modalities, and rely on latent variables for generation. For experimental design or creative applications that require more control over the generation process, it has become common to manipulate the latent variable directly. However, existing approaches for performing such manipulations (e.g. interpolation or forming low-dimensional representations) only work well in special cases or are network or data-modality specific. We propose Latent Optimal Linear combinations (LOL) as a general-purpose method to form linear combinations of latent variables that adhere to the assumptions of the generative model. As LOL is easy to implement and naturally addresses the broader task of forming any linear combinations, e.g. the construction of subspaces of the latent space, LOL dramatically simplifies the creation of expressive low-dimensional representations of high-dimensional objects.

• The paper proposes COG, a method that transforms linear combinations of latents to preserve Gaussian distributions for diffusion models.
• It demonstrates superior performance over baseline interpolation techniques with improved FID scores and accuracy in centroid determination.
• The method enhances latent space manipulation, enabling precise interpolation and projection for high-quality generative modeling.

Linear Combinations of Latents in Diffusion Models: Interpolation and Beyond

Linear combinations of latents in diffusion models authored by Erik Bodin, Henry Moss, and Carl Henrik Ek, addresses a critical challenge in generative modeling: the control and manipulation of latent variables. Generative models, notably diffusion models, flow matching, and continuous normalizing flows, have shown substantial effectiveness across various modalities. However, current methods of combining latent variables, such as spherical interpolation, are limited to special cases and are often not generalizable. This paper proposes a novel method, Combination of Gaussian variables (COG), which ensures that interpolated latents follow the distribution expected by the generative model.

Core Contributions and Methodology

The essence of this work is the insight that standard interpolation methods fail because they do not preserve the distribution on which generative models like diffusion and flow matching are trained. Specifically, simple linear interpolation does not guarantee that interpolated points follow the Gaussian distribution these models expect, often resulting in poor generation quality. Recognizing this, the authors leverage the Gaussian distribution properties and introduce a method to ensure that any linear combination of latents results in valid Gaussian-distributed vectors, applicable across various operations such as interpolation, centroid determination, and subspace projections.

COG achieves this by transforming linear combinations of latents to match the predefined Gaussian distribution through a closed-form expression:

$\bm{z} = \bm{a} + \bm{B} \bm{y}$

where $\bm{a}$ and $\bm{B}$ are defined to reweight the latent vectors to ensure that the transformed random variable $\bm{z}$ adheres to the distribution $\mathcal{N}(\bm{\mu}, \bm{\Sigma})$.

Experimental Verification

The authors conduct rigorous experimental comparisons to demonstrate the efficacy of their proposed method. They compare COG with baseline techniques: linear interpolation (LERP), spherical linear interpolation (SLERP), and the Norm-Aware Optimization (NAO) method across two key applications: interpolation and centroid determination. The experiments utilize Stable Diffusion (SD) 2.1 and ImageNet dataset, applying quantitative metrics like FID scores and color classification accuracy using a pre-trained classifier.

Interpolation Results

For interpolation, the paper presents compelling numerical results where COG outperforms state-of-the-art methods:

• Accuracy: COG achieved 67.39%, outperforming NAO at 62.13%.
• FID Score: With a score of 38.87, COG surpassed both SLERP and NAO.

The results indicate that COG produces more visually coherent interpolations with higher semantic preservation between endpoints.

Centroid Determination

For centroid determination, the comparison against baselines was similarly favorable:

• Accuracy: COG achieved 46.29%, higher than NAO at 44.00%.
• FID Scores: COG showed competitive advantages here as well, establishing it as an effective tool for centroid determination.

The paper corroborates these numeric results with qualitative illustrations showcasing the visual enhancements achieved through COG, particularly noting the removal of artifacts and improved semantic consistency in generated images.

Implications and Future Directions

The contributions of COG extend beyond simple interpolation, demonstrating its versatility for general linear combinations and subspace projections. This flexibility enables the construction of meaningful low-dimensional representations from high-dimensional latent spaces, a crucial advancement for applications in creative generation and surrogate modeling.

The implications of this research are significant. The ability to accurately manipulate latent spaces can dramatically improve the control and quality of generated content across various tasks in image synthesis, video creation, and potentially 3D modeling. These findings open new avenues for research, particularly in enhancing the robustness of generative models and exploring even more complex transformations within latent spaces.

Conclusion

Erik Bodin, Henry Moss, and Carl Henrik Ek's paper successfully addresses the limitations in current generative modeling techniques by proposing a universally applicable method for combining latents. COG stands out for its simplicity, theoretical rigor, and practical effectiveness in improving the manipulation of latent spaces, making a substantive contribution to the field of generative modeling. This work lays a foundation for future developments that could explore even broader applications and optimizations within AI-driven generative processes.