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

arXiv: https://arxiv.org/abs/2511.12757 · HTML: https://mlsanity.com/paper/2511.12757/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 32 of its 50 references that are in the map.
- First posted on arXiv 16 Nov 2025.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 32)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [An Image is Worth One Word: Personalizing Text-to-Image Generation using Textual Inversion](https://mlsanity.com/paper/2208.01618/) (2022)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Ablating Concepts in Text-to-Image Diffusion Models](https://mlsanity.com/paper/2303.13516/) (2023)
- [DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking](https://mlsanity.com/paper/2210.01776/) (2022)
- [Unsupervised Discovery of Interpretable Directions in the GAN Latent Space](https://mlsanity.com/paper/2002.03754/) (2020)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2307.12868/) (2023)
- [Diffusion Models Beat GANs on Topology Optimization](https://mlsanity.com/paper/2208.09591/) (2022)
- [DiffMorpher: Unleashing the Capability of Diffusion Models for Image Morphing](https://mlsanity.com/paper/2312.07409/) (2023)
- [Get What You Want, Not What You Don't: Image Content Suppression for Text-to-Image Diffusion Models](https://mlsanity.com/paper/2402.05375/) (2024)
- [Mad Max: Affine Spline Insights into Deep Learning](https://mlsanity.com/paper/1805.06576/) (2018)
- [Crisscrossed Captions: Extended Intramodal and Intermodal Semantic Similarity Judgments for MS-COCO](https://mlsanity.com/paper/2004.15020/) (2020)
- [Interpolating between Images with Diffusion Models](https://mlsanity.com/paper/2307.12560/) (2023)
- [On Discrete Prompt Optimization for Diffusion Models](https://mlsanity.com/paper/2407.01606/) (2024)
- [AID: Attention Interpolation of Text-to-Image Diffusion](https://mlsanity.com/paper/2403.17924/) (2024)
- [IMPUS: Image Morphing with Perceptually-Uniform Sampling Using Diffusion Models](https://mlsanity.com/paper/2311.06792/) (2023)
- [Compositional Inversion for Stable Diffusion Models](https://mlsanity.com/paper/2312.08048/) (2023)
- [TVG: A Training-free Transition Video Generation Method with Diffusion Models](https://mlsanity.com/paper/2408.13413/) (2024)
- [Geometric Uncertainty for Detecting and Correcting Hallucinations in LLMs](https://mlsanity.com/paper/2509.13813/) (2025)

## Built on by (most cited first, 1 of 1)

- [Consistency Regularised Gradient Flows for Inverse Problems](https://mlsanity.com/paper/2605.07907/) (2026)

## Abstract (arXiv)

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.