Diffusion Paths: A Geodesic Lens on Noising Schedules
Sahil Akhtar · Team Sahil
Submitted to AI Safety x Physics Grand Challenge. Sprint projects are early-stage work by participants, not Apart Research publications.
We examine whether there is a theoretical basis for the choices of noising schedules in diffusion models. We look at this by considering the noising schedule as probability path from the initial distribution to the gaussian noise. We examine if adherence to geodesic nature means a better noising schedule in terms of sample quality and epochs to learn.
Reviews
While of limited applicability to AI safety, the experimental harness is clean and the results are relatively clear. However, while there is a story to tell here (i.e that the linear schedule is optimal) there could be more substantial discussion and explanation as to why this might be. Furthermore, experiments on non-toy datasets are required as it's unclear why this result should generalize beyond this restricted domain.
Good hackathon project. The problem seems crisp, the result is somewhat surprising, the toy dataset is reasonable and visually satisfying. I would have liked a baseline comparison with a (perhaps suitably whitened) image dataset and a bit more discussion of path energy, how it is expected to affect efficiency, and what could be causing the difference in this case.
Interpreting noise schedules through a geodesic lens is a clear, physics‑grounded idea. The experimental scope is modest (toy 2D GMM), but the setup is clean and reproducible in spirit. The main limitation is the distance from concrete AI‑safety questions; the work is primarily a training/optimization study for diffusion models, with only indirect safety implications. Strengthening the theoretical link between conditional and marginal geodesics, scaling to higher‑dimensional data, and adding safety‑centric evaluations would materially increase impact. But I found the thought behind the paper intersting. Some notes on how I graded it: Criteria1 : obviously clean and simplistic setup, well written, code works but conclusions about optimality are likely to not hold for real models. 2. Sorry but its more of optimisation problem, maybe interp related so I put 2 here. 3. Here its clearly new, physics driven idea, really nice
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The project investigates the effect of the noising schedule on diffusion model performance, and investigates a novel schedule based on the Wasserstein-2 geodesic. The project is well executed and the report is clearly written. However, the project does not appear to address AI safety at all, which is a crucial shortcoming.
Cite this project
@misc{akhtar2025diffusion,
title = {{Diffusion Paths: A Geodesic Lens on Noising Schedules}},
author = {Sahil Akhtar},
year = {2025},
month = jul,
note = {Submitted to AI Safety x Physics Grand Challenge, an Apart Research Sprint},
howpublished = {\url{https://apartresearch.com/sprints/projects/diffusion-paths-a-geodesic-lens-on-noising-schedules-rpaq}},
url = {https://apartresearch.com/sprints/projects/diffusion-paths-a-geodesic-lens-on-noising-schedules-rpaq}
}More from AI Safety x Physics Grand Challenge
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