Which Way from B to A: The role of embedding geometry in image interpolation for Stable Diffusion

TL;DR

This paper introduces a novel approach to image interpolation in Stable Diffusion by interpreting CLIP embeddings as point clouds in Wasserstein space and using optimal transport to produce smoother, more coherent intermediate images.

Contribution

It proposes a new geometric perspective on CLIP embeddings and applies optimal transport to improve image interpolation quality in generative models.

Findings

Optimal transport-based interpolation yields smoother images.

Viewing embeddings as point clouds enhances geometric understanding.

The method outperforms standard interpolation techniques.

Abstract

It can be shown that Stable Diffusion has a permutation-invariance property with respect to the rows of Contrastive Language-Image Pretraining (CLIP) embedding matrices. This inspired the novel observation that these embeddings can naturally be interpreted as point clouds in a Wasserstein space rather than as matrices in a Euclidean space. This perspective opens up new possibilities for understanding the geometry of embedding space. For example, when interpolating between embeddings of two distinct prompts, we propose reframing the interpolation problem as an optimal transport problem. By solving this optimal transport problem, we compute a shortest path (or geodesic) between embeddings that captures a more natural and geometrically smooth transition through the embedding space. This results in smoother and more coherent intermediate (interpolated) images when rendered by the Stable Diffusion generative model. We conduct experiments to investigate this effect, comparing the quality of interpolated images produced using optimal transport to those generated by other standard interpolation methods. The novel optimal transport--based approach presented indeed gives smoother image interpolations, suggesting that viewing the embeddings as point clouds (rather than as matrices) better reflects and leverages the geometry of the embedding space.