Introduction
Analyzing the latent space of diffusion models (DMs) from a geometrical perspective can provide us with insight into the underlying structure and relationships encoded within the data. By examining the latent space of DMs through a geometrical perspective, we can uncover patterns, clusters, and representations that may not be immediately apparent in the raw data. This approach enables us to understand the intrinsic geometry of the data manifold, revealing its high-dimensional structure in a more interpretable manner. Additionally, analyzing the latent space geometrically can facilitate tasks such as dimensionality reduction, visualization, and clustering, thereby aiding in model interpretation, feature extraction, and downstream analysis. Leverage the concept of a pullback metric from Riemannian geometry, one can understand the local latent basis and the corresponding local tangent basis in the feature space as elucidated in Park et al. (2024).
Local Latent Basis
The main idea is to discover the local latent basis in the latent space \(X\), which is achieved by performing singular value decomposition (SVD) on the Jacobian matrix \(J_x = \nabla_x h\) of the map \(f: X \rightarrow H\), where \(h\) represents the bottleneck representation in \(H\), and \(H\) is the feature space associated with the bottleneck layer of the U-Net architecture employed in the diffusion model. The SVD yields \(J_x = U \Lambda V^T\), where the columns of \(V\) represent the local latent basis vectors \(\{v_1, v_2, \ldots, v_n\}\) in \(X\), and the columns of \(U\) represent the corresponding local tangent basis vectors \(\{u_1, u_2, \ldots, u_n\}\) in \(H\). These basis vectors capture the directions of maximal variability in the latent and feature spaces, respectively. In addition, they enable semantically meaningful image editing by moving along the basis vector at specific timesteps. By converting the latent direction \(v_i \in T_x\) to the corresponding direction \(u_i \in T_h\), applying parallel transport to move \(u_i\) to \(u'_i \in T_{h'}\), and transforming \(u'_i\) back to \(v'_i \in T_{x'}\), it is shown that the editing directions can be consistently applied across different samples without the need for manual identification of semantic relevance for each sample.
Generative Process
Park et al. investigate how the geometric structure of DMs evolves during the generative process and differs across different text conditions. They observe that the frequency domain of the local latent basis shifts from low-frequency to high-frequency along the generative process, and the discrepancy between local tangent spaces of different samples increases along the generative process, indicating that finding universally applicable editing directions becomes more challenging at later timesteps.
Moreover, the study explores the impact of conditioning prompts on the latent structure of text-to-image diffusion models (DMs). It finds that local tangent spaces for semantically similar prompts exhibit substantial overlap, while those for semantically distant prompts show significant variation, providing insights into the model's understanding of prompt semantics and its ability to capture fine-grained distinctions between different prompts.
Application in Image Editing
Building upon these observations, Park et al. introduce a novel approach to semantic image editing in diffusion models (DMs). They propose that by leveraging the local latent basis, one can perform semantically meaningful image editing by moving along the basis vector at specific timesteps. This enables modifications such as changing the object's color, shape, and texture, without requiring manual identification of semantic relevance for each sample. They demonstrate that their approach can be applied to various applications, including image manipulation, style transfer, and data augmentation, achieving high-quality and semantically consistent results.
Conclusion
The integration of Riemannian geometry and diffusion models (DMs) offers a promising direction for understanding and improving the interpretability and functionality of these models. By leveraging the pullback metric and local latent basis, researchers can uncover the intrinsic geometry of the data manifold, leading to more interpretable and semantically meaningful representations. Furthermore, this approach enables novel applications in image editing, style transfer, and data augmentation, highlighting the potential for Riemannian geometry to advance the field of diffusion models and their practical utility.